quantum dots by timothy paik marcus dahlstrom michael nip
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Quantum Dots
By Timothy Paik
Marcus DahlstromMichael Nip
Implementing Quantum Computers
bull Many implementations for quantum computing
bull Why solid statendash Scalabilityndash Decoherence is less of a problem
What is a quantum dot
bull In two words a semiconductor nanocrystal
bull Easily tunable by changing the size and composition of the nanocrystal
Gallium Arsenide Quantum Dots
bull Gallium arsenide is a III-V semiconductorndash Higher saturated electron velocity and higher
electron mobility than siliconndash Gallium arsenide can emit and absorb light
unlike siliconbull No silicon laser is possible (or has been made yet)
Energy Band Levels
bull Electrons exist in discrete energy levels in bulk semiconductor materialndash There exists a forbidden
range of energy levels in any material called the band gap
Energy Band Levels
bull By absorbing some sort of stimulus (in light or heat form) an electron can rise to the conduction band from the valence bandndash This action leaves behind a
ldquoholerdquo in the valence band The hole and the electron together are called an exciton
Energy Band Levels
bull The average distance between an electron and a hole in a exciton is called the Excited Bohr Radius
bull When the size of the semiconductor falls below the Bohr Radius the semiconductor is called a quantum dot
Tuning Quantum Dots
bull By changing size shape and composition quantum dots can change their absorptive and emissive properties dramatically
Manufacturing methods
bull Electron beam lithography
bull Molecular beam epitaxy
Electron Beam Lithography
bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer
bull λ = (123 Aring radicV)bull Advantages
ndash generation of micron and submicron resist geometries
ndash greater depth of focus than optical lithography
ndash masks are unnecessaryndash Optical diffraction limit is not a
real concern
Electron Beam Lithography
bull Disadvantage(s)ndash The lithography is serial
(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution
ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution
bull Heavy ion lithography has been proposed but still is in development stages
Molecular Beam Epitaxy
bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to
combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms
only meet each other on the wafer
Spin Quantum Computing
Qubit information is stored in the spin state of an electron in an artificial atom
AdvantagesLong decoherence time
Future Scalabilty
Artifical atoms are bigger than regular atoms therefore easier to manipulate
Decoherence time ~ 100ns
bull Time before the quantum mechanical system starts acting in a classical way with its complex environment
bull The state of the system has not yet collapsed due to (unwanted) environmental effects
bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error
correction This requirement is met
Artificial Atombull Double Barrier
Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength
rarr Discrete energy levels
Adding Electrons changing Vgate
bull 2D-Harmonic Oscillator
bull Shell structure as in atoms
bull Magic Numbers 2 6 12
bull To add ldquoevenrdquo electron requires only additional Coulomb energy
Comparison with Hydrogenbull Artificial Atom
Energy levels ~ 1meV
Size ~ 10μm
Weak magnetic fields can affect energy levels
bull Hydrogen
Energy levels ~ 1eV
Size ~ 1Aring
Only strong magnetic fields can perturb energy levels
Factor 1000
Tuning the Quantum Dot
bull Tune so we have one valence electron
bull Initial state can be set by applying homogeneous magnetic field rarr |0gt
bull Low temperature kT lt ΔE (state gap)
bull Now we have defined our single qubit
Energy
position
Gate bias
Spin up - electron
Unoccupied state
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Implementing Quantum Computers
bull Many implementations for quantum computing
bull Why solid statendash Scalabilityndash Decoherence is less of a problem
What is a quantum dot
bull In two words a semiconductor nanocrystal
bull Easily tunable by changing the size and composition of the nanocrystal
Gallium Arsenide Quantum Dots
bull Gallium arsenide is a III-V semiconductorndash Higher saturated electron velocity and higher
electron mobility than siliconndash Gallium arsenide can emit and absorb light
unlike siliconbull No silicon laser is possible (or has been made yet)
Energy Band Levels
bull Electrons exist in discrete energy levels in bulk semiconductor materialndash There exists a forbidden
range of energy levels in any material called the band gap
Energy Band Levels
bull By absorbing some sort of stimulus (in light or heat form) an electron can rise to the conduction band from the valence bandndash This action leaves behind a
ldquoholerdquo in the valence band The hole and the electron together are called an exciton
Energy Band Levels
bull The average distance between an electron and a hole in a exciton is called the Excited Bohr Radius
bull When the size of the semiconductor falls below the Bohr Radius the semiconductor is called a quantum dot
Tuning Quantum Dots
bull By changing size shape and composition quantum dots can change their absorptive and emissive properties dramatically
Manufacturing methods
bull Electron beam lithography
bull Molecular beam epitaxy
Electron Beam Lithography
bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer
bull λ = (123 Aring radicV)bull Advantages
ndash generation of micron and submicron resist geometries
ndash greater depth of focus than optical lithography
ndash masks are unnecessaryndash Optical diffraction limit is not a
real concern
Electron Beam Lithography
bull Disadvantage(s)ndash The lithography is serial
(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution
ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution
bull Heavy ion lithography has been proposed but still is in development stages
Molecular Beam Epitaxy
bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to
combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms
only meet each other on the wafer
Spin Quantum Computing
Qubit information is stored in the spin state of an electron in an artificial atom
AdvantagesLong decoherence time
Future Scalabilty
Artifical atoms are bigger than regular atoms therefore easier to manipulate
Decoherence time ~ 100ns
bull Time before the quantum mechanical system starts acting in a classical way with its complex environment
bull The state of the system has not yet collapsed due to (unwanted) environmental effects
bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error
correction This requirement is met
Artificial Atombull Double Barrier
Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength
rarr Discrete energy levels
Adding Electrons changing Vgate
bull 2D-Harmonic Oscillator
bull Shell structure as in atoms
bull Magic Numbers 2 6 12
bull To add ldquoevenrdquo electron requires only additional Coulomb energy
Comparison with Hydrogenbull Artificial Atom
Energy levels ~ 1meV
Size ~ 10μm
Weak magnetic fields can affect energy levels
bull Hydrogen
Energy levels ~ 1eV
Size ~ 1Aring
Only strong magnetic fields can perturb energy levels
Factor 1000
Tuning the Quantum Dot
bull Tune so we have one valence electron
bull Initial state can be set by applying homogeneous magnetic field rarr |0gt
bull Low temperature kT lt ΔE (state gap)
bull Now we have defined our single qubit
Energy
position
Gate bias
Spin up - electron
Unoccupied state
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
What is a quantum dot
bull In two words a semiconductor nanocrystal
bull Easily tunable by changing the size and composition of the nanocrystal
Gallium Arsenide Quantum Dots
bull Gallium arsenide is a III-V semiconductorndash Higher saturated electron velocity and higher
electron mobility than siliconndash Gallium arsenide can emit and absorb light
unlike siliconbull No silicon laser is possible (or has been made yet)
Energy Band Levels
bull Electrons exist in discrete energy levels in bulk semiconductor materialndash There exists a forbidden
range of energy levels in any material called the band gap
Energy Band Levels
bull By absorbing some sort of stimulus (in light or heat form) an electron can rise to the conduction band from the valence bandndash This action leaves behind a
ldquoholerdquo in the valence band The hole and the electron together are called an exciton
Energy Band Levels
bull The average distance between an electron and a hole in a exciton is called the Excited Bohr Radius
bull When the size of the semiconductor falls below the Bohr Radius the semiconductor is called a quantum dot
Tuning Quantum Dots
bull By changing size shape and composition quantum dots can change their absorptive and emissive properties dramatically
Manufacturing methods
bull Electron beam lithography
bull Molecular beam epitaxy
Electron Beam Lithography
bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer
bull λ = (123 Aring radicV)bull Advantages
ndash generation of micron and submicron resist geometries
ndash greater depth of focus than optical lithography
ndash masks are unnecessaryndash Optical diffraction limit is not a
real concern
Electron Beam Lithography
bull Disadvantage(s)ndash The lithography is serial
(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution
ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution
bull Heavy ion lithography has been proposed but still is in development stages
Molecular Beam Epitaxy
bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to
combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms
only meet each other on the wafer
Spin Quantum Computing
Qubit information is stored in the spin state of an electron in an artificial atom
AdvantagesLong decoherence time
Future Scalabilty
Artifical atoms are bigger than regular atoms therefore easier to manipulate
Decoherence time ~ 100ns
bull Time before the quantum mechanical system starts acting in a classical way with its complex environment
bull The state of the system has not yet collapsed due to (unwanted) environmental effects
bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error
correction This requirement is met
Artificial Atombull Double Barrier
Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength
rarr Discrete energy levels
Adding Electrons changing Vgate
bull 2D-Harmonic Oscillator
bull Shell structure as in atoms
bull Magic Numbers 2 6 12
bull To add ldquoevenrdquo electron requires only additional Coulomb energy
Comparison with Hydrogenbull Artificial Atom
Energy levels ~ 1meV
Size ~ 10μm
Weak magnetic fields can affect energy levels
bull Hydrogen
Energy levels ~ 1eV
Size ~ 1Aring
Only strong magnetic fields can perturb energy levels
Factor 1000
Tuning the Quantum Dot
bull Tune so we have one valence electron
bull Initial state can be set by applying homogeneous magnetic field rarr |0gt
bull Low temperature kT lt ΔE (state gap)
bull Now we have defined our single qubit
Energy
position
Gate bias
Spin up - electron
Unoccupied state
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Gallium Arsenide Quantum Dots
bull Gallium arsenide is a III-V semiconductorndash Higher saturated electron velocity and higher
electron mobility than siliconndash Gallium arsenide can emit and absorb light
unlike siliconbull No silicon laser is possible (or has been made yet)
Energy Band Levels
bull Electrons exist in discrete energy levels in bulk semiconductor materialndash There exists a forbidden
range of energy levels in any material called the band gap
Energy Band Levels
bull By absorbing some sort of stimulus (in light or heat form) an electron can rise to the conduction band from the valence bandndash This action leaves behind a
ldquoholerdquo in the valence band The hole and the electron together are called an exciton
Energy Band Levels
bull The average distance between an electron and a hole in a exciton is called the Excited Bohr Radius
bull When the size of the semiconductor falls below the Bohr Radius the semiconductor is called a quantum dot
Tuning Quantum Dots
bull By changing size shape and composition quantum dots can change their absorptive and emissive properties dramatically
Manufacturing methods
bull Electron beam lithography
bull Molecular beam epitaxy
Electron Beam Lithography
bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer
bull λ = (123 Aring radicV)bull Advantages
ndash generation of micron and submicron resist geometries
ndash greater depth of focus than optical lithography
ndash masks are unnecessaryndash Optical diffraction limit is not a
real concern
Electron Beam Lithography
bull Disadvantage(s)ndash The lithography is serial
(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution
ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution
bull Heavy ion lithography has been proposed but still is in development stages
Molecular Beam Epitaxy
bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to
combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms
only meet each other on the wafer
Spin Quantum Computing
Qubit information is stored in the spin state of an electron in an artificial atom
AdvantagesLong decoherence time
Future Scalabilty
Artifical atoms are bigger than regular atoms therefore easier to manipulate
Decoherence time ~ 100ns
bull Time before the quantum mechanical system starts acting in a classical way with its complex environment
bull The state of the system has not yet collapsed due to (unwanted) environmental effects
bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error
correction This requirement is met
Artificial Atombull Double Barrier
Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength
rarr Discrete energy levels
Adding Electrons changing Vgate
bull 2D-Harmonic Oscillator
bull Shell structure as in atoms
bull Magic Numbers 2 6 12
bull To add ldquoevenrdquo electron requires only additional Coulomb energy
Comparison with Hydrogenbull Artificial Atom
Energy levels ~ 1meV
Size ~ 10μm
Weak magnetic fields can affect energy levels
bull Hydrogen
Energy levels ~ 1eV
Size ~ 1Aring
Only strong magnetic fields can perturb energy levels
Factor 1000
Tuning the Quantum Dot
bull Tune so we have one valence electron
bull Initial state can be set by applying homogeneous magnetic field rarr |0gt
bull Low temperature kT lt ΔE (state gap)
bull Now we have defined our single qubit
Energy
position
Gate bias
Spin up - electron
Unoccupied state
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Energy Band Levels
bull Electrons exist in discrete energy levels in bulk semiconductor materialndash There exists a forbidden
range of energy levels in any material called the band gap
Energy Band Levels
bull By absorbing some sort of stimulus (in light or heat form) an electron can rise to the conduction band from the valence bandndash This action leaves behind a
ldquoholerdquo in the valence band The hole and the electron together are called an exciton
Energy Band Levels
bull The average distance between an electron and a hole in a exciton is called the Excited Bohr Radius
bull When the size of the semiconductor falls below the Bohr Radius the semiconductor is called a quantum dot
Tuning Quantum Dots
bull By changing size shape and composition quantum dots can change their absorptive and emissive properties dramatically
Manufacturing methods
bull Electron beam lithography
bull Molecular beam epitaxy
Electron Beam Lithography
bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer
bull λ = (123 Aring radicV)bull Advantages
ndash generation of micron and submicron resist geometries
ndash greater depth of focus than optical lithography
ndash masks are unnecessaryndash Optical diffraction limit is not a
real concern
Electron Beam Lithography
bull Disadvantage(s)ndash The lithography is serial
(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution
ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution
bull Heavy ion lithography has been proposed but still is in development stages
Molecular Beam Epitaxy
bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to
combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms
only meet each other on the wafer
Spin Quantum Computing
Qubit information is stored in the spin state of an electron in an artificial atom
AdvantagesLong decoherence time
Future Scalabilty
Artifical atoms are bigger than regular atoms therefore easier to manipulate
Decoherence time ~ 100ns
bull Time before the quantum mechanical system starts acting in a classical way with its complex environment
bull The state of the system has not yet collapsed due to (unwanted) environmental effects
bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error
correction This requirement is met
Artificial Atombull Double Barrier
Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength
rarr Discrete energy levels
Adding Electrons changing Vgate
bull 2D-Harmonic Oscillator
bull Shell structure as in atoms
bull Magic Numbers 2 6 12
bull To add ldquoevenrdquo electron requires only additional Coulomb energy
Comparison with Hydrogenbull Artificial Atom
Energy levels ~ 1meV
Size ~ 10μm
Weak magnetic fields can affect energy levels
bull Hydrogen
Energy levels ~ 1eV
Size ~ 1Aring
Only strong magnetic fields can perturb energy levels
Factor 1000
Tuning the Quantum Dot
bull Tune so we have one valence electron
bull Initial state can be set by applying homogeneous magnetic field rarr |0gt
bull Low temperature kT lt ΔE (state gap)
bull Now we have defined our single qubit
Energy
position
Gate bias
Spin up - electron
Unoccupied state
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Energy Band Levels
bull By absorbing some sort of stimulus (in light or heat form) an electron can rise to the conduction band from the valence bandndash This action leaves behind a
ldquoholerdquo in the valence band The hole and the electron together are called an exciton
Energy Band Levels
bull The average distance between an electron and a hole in a exciton is called the Excited Bohr Radius
bull When the size of the semiconductor falls below the Bohr Radius the semiconductor is called a quantum dot
Tuning Quantum Dots
bull By changing size shape and composition quantum dots can change their absorptive and emissive properties dramatically
Manufacturing methods
bull Electron beam lithography
bull Molecular beam epitaxy
Electron Beam Lithography
bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer
bull λ = (123 Aring radicV)bull Advantages
ndash generation of micron and submicron resist geometries
ndash greater depth of focus than optical lithography
ndash masks are unnecessaryndash Optical diffraction limit is not a
real concern
Electron Beam Lithography
bull Disadvantage(s)ndash The lithography is serial
(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution
ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution
bull Heavy ion lithography has been proposed but still is in development stages
Molecular Beam Epitaxy
bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to
combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms
only meet each other on the wafer
Spin Quantum Computing
Qubit information is stored in the spin state of an electron in an artificial atom
AdvantagesLong decoherence time
Future Scalabilty
Artifical atoms are bigger than regular atoms therefore easier to manipulate
Decoherence time ~ 100ns
bull Time before the quantum mechanical system starts acting in a classical way with its complex environment
bull The state of the system has not yet collapsed due to (unwanted) environmental effects
bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error
correction This requirement is met
Artificial Atombull Double Barrier
Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength
rarr Discrete energy levels
Adding Electrons changing Vgate
bull 2D-Harmonic Oscillator
bull Shell structure as in atoms
bull Magic Numbers 2 6 12
bull To add ldquoevenrdquo electron requires only additional Coulomb energy
Comparison with Hydrogenbull Artificial Atom
Energy levels ~ 1meV
Size ~ 10μm
Weak magnetic fields can affect energy levels
bull Hydrogen
Energy levels ~ 1eV
Size ~ 1Aring
Only strong magnetic fields can perturb energy levels
Factor 1000
Tuning the Quantum Dot
bull Tune so we have one valence electron
bull Initial state can be set by applying homogeneous magnetic field rarr |0gt
bull Low temperature kT lt ΔE (state gap)
bull Now we have defined our single qubit
Energy
position
Gate bias
Spin up - electron
Unoccupied state
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Energy Band Levels
bull The average distance between an electron and a hole in a exciton is called the Excited Bohr Radius
bull When the size of the semiconductor falls below the Bohr Radius the semiconductor is called a quantum dot
Tuning Quantum Dots
bull By changing size shape and composition quantum dots can change their absorptive and emissive properties dramatically
Manufacturing methods
bull Electron beam lithography
bull Molecular beam epitaxy
Electron Beam Lithography
bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer
bull λ = (123 Aring radicV)bull Advantages
ndash generation of micron and submicron resist geometries
ndash greater depth of focus than optical lithography
ndash masks are unnecessaryndash Optical diffraction limit is not a
real concern
Electron Beam Lithography
bull Disadvantage(s)ndash The lithography is serial
(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution
ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution
bull Heavy ion lithography has been proposed but still is in development stages
Molecular Beam Epitaxy
bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to
combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms
only meet each other on the wafer
Spin Quantum Computing
Qubit information is stored in the spin state of an electron in an artificial atom
AdvantagesLong decoherence time
Future Scalabilty
Artifical atoms are bigger than regular atoms therefore easier to manipulate
Decoherence time ~ 100ns
bull Time before the quantum mechanical system starts acting in a classical way with its complex environment
bull The state of the system has not yet collapsed due to (unwanted) environmental effects
bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error
correction This requirement is met
Artificial Atombull Double Barrier
Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength
rarr Discrete energy levels
Adding Electrons changing Vgate
bull 2D-Harmonic Oscillator
bull Shell structure as in atoms
bull Magic Numbers 2 6 12
bull To add ldquoevenrdquo electron requires only additional Coulomb energy
Comparison with Hydrogenbull Artificial Atom
Energy levels ~ 1meV
Size ~ 10μm
Weak magnetic fields can affect energy levels
bull Hydrogen
Energy levels ~ 1eV
Size ~ 1Aring
Only strong magnetic fields can perturb energy levels
Factor 1000
Tuning the Quantum Dot
bull Tune so we have one valence electron
bull Initial state can be set by applying homogeneous magnetic field rarr |0gt
bull Low temperature kT lt ΔE (state gap)
bull Now we have defined our single qubit
Energy
position
Gate bias
Spin up - electron
Unoccupied state
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Tuning Quantum Dots
bull By changing size shape and composition quantum dots can change their absorptive and emissive properties dramatically
Manufacturing methods
bull Electron beam lithography
bull Molecular beam epitaxy
Electron Beam Lithography
bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer
bull λ = (123 Aring radicV)bull Advantages
ndash generation of micron and submicron resist geometries
ndash greater depth of focus than optical lithography
ndash masks are unnecessaryndash Optical diffraction limit is not a
real concern
Electron Beam Lithography
bull Disadvantage(s)ndash The lithography is serial
(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution
ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution
bull Heavy ion lithography has been proposed but still is in development stages
Molecular Beam Epitaxy
bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to
combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms
only meet each other on the wafer
Spin Quantum Computing
Qubit information is stored in the spin state of an electron in an artificial atom
AdvantagesLong decoherence time
Future Scalabilty
Artifical atoms are bigger than regular atoms therefore easier to manipulate
Decoherence time ~ 100ns
bull Time before the quantum mechanical system starts acting in a classical way with its complex environment
bull The state of the system has not yet collapsed due to (unwanted) environmental effects
bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error
correction This requirement is met
Artificial Atombull Double Barrier
Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength
rarr Discrete energy levels
Adding Electrons changing Vgate
bull 2D-Harmonic Oscillator
bull Shell structure as in atoms
bull Magic Numbers 2 6 12
bull To add ldquoevenrdquo electron requires only additional Coulomb energy
Comparison with Hydrogenbull Artificial Atom
Energy levels ~ 1meV
Size ~ 10μm
Weak magnetic fields can affect energy levels
bull Hydrogen
Energy levels ~ 1eV
Size ~ 1Aring
Only strong magnetic fields can perturb energy levels
Factor 1000
Tuning the Quantum Dot
bull Tune so we have one valence electron
bull Initial state can be set by applying homogeneous magnetic field rarr |0gt
bull Low temperature kT lt ΔE (state gap)
bull Now we have defined our single qubit
Energy
position
Gate bias
Spin up - electron
Unoccupied state
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Manufacturing methods
bull Electron beam lithography
bull Molecular beam epitaxy
Electron Beam Lithography
bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer
bull λ = (123 Aring radicV)bull Advantages
ndash generation of micron and submicron resist geometries
ndash greater depth of focus than optical lithography
ndash masks are unnecessaryndash Optical diffraction limit is not a
real concern
Electron Beam Lithography
bull Disadvantage(s)ndash The lithography is serial
(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution
ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution
bull Heavy ion lithography has been proposed but still is in development stages
Molecular Beam Epitaxy
bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to
combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms
only meet each other on the wafer
Spin Quantum Computing
Qubit information is stored in the spin state of an electron in an artificial atom
AdvantagesLong decoherence time
Future Scalabilty
Artifical atoms are bigger than regular atoms therefore easier to manipulate
Decoherence time ~ 100ns
bull Time before the quantum mechanical system starts acting in a classical way with its complex environment
bull The state of the system has not yet collapsed due to (unwanted) environmental effects
bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error
correction This requirement is met
Artificial Atombull Double Barrier
Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength
rarr Discrete energy levels
Adding Electrons changing Vgate
bull 2D-Harmonic Oscillator
bull Shell structure as in atoms
bull Magic Numbers 2 6 12
bull To add ldquoevenrdquo electron requires only additional Coulomb energy
Comparison with Hydrogenbull Artificial Atom
Energy levels ~ 1meV
Size ~ 10μm
Weak magnetic fields can affect energy levels
bull Hydrogen
Energy levels ~ 1eV
Size ~ 1Aring
Only strong magnetic fields can perturb energy levels
Factor 1000
Tuning the Quantum Dot
bull Tune so we have one valence electron
bull Initial state can be set by applying homogeneous magnetic field rarr |0gt
bull Low temperature kT lt ΔE (state gap)
bull Now we have defined our single qubit
Energy
position
Gate bias
Spin up - electron
Unoccupied state
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Electron Beam Lithography
bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer
bull λ = (123 Aring radicV)bull Advantages
ndash generation of micron and submicron resist geometries
ndash greater depth of focus than optical lithography
ndash masks are unnecessaryndash Optical diffraction limit is not a
real concern
Electron Beam Lithography
bull Disadvantage(s)ndash The lithography is serial
(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution
ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution
bull Heavy ion lithography has been proposed but still is in development stages
Molecular Beam Epitaxy
bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to
combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms
only meet each other on the wafer
Spin Quantum Computing
Qubit information is stored in the spin state of an electron in an artificial atom
AdvantagesLong decoherence time
Future Scalabilty
Artifical atoms are bigger than regular atoms therefore easier to manipulate
Decoherence time ~ 100ns
bull Time before the quantum mechanical system starts acting in a classical way with its complex environment
bull The state of the system has not yet collapsed due to (unwanted) environmental effects
bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error
correction This requirement is met
Artificial Atombull Double Barrier
Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength
rarr Discrete energy levels
Adding Electrons changing Vgate
bull 2D-Harmonic Oscillator
bull Shell structure as in atoms
bull Magic Numbers 2 6 12
bull To add ldquoevenrdquo electron requires only additional Coulomb energy
Comparison with Hydrogenbull Artificial Atom
Energy levels ~ 1meV
Size ~ 10μm
Weak magnetic fields can affect energy levels
bull Hydrogen
Energy levels ~ 1eV
Size ~ 1Aring
Only strong magnetic fields can perturb energy levels
Factor 1000
Tuning the Quantum Dot
bull Tune so we have one valence electron
bull Initial state can be set by applying homogeneous magnetic field rarr |0gt
bull Low temperature kT lt ΔE (state gap)
bull Now we have defined our single qubit
Energy
position
Gate bias
Spin up - electron
Unoccupied state
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Electron Beam Lithography
bull Disadvantage(s)ndash The lithography is serial
(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution
ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution
bull Heavy ion lithography has been proposed but still is in development stages
Molecular Beam Epitaxy
bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to
combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms
only meet each other on the wafer
Spin Quantum Computing
Qubit information is stored in the spin state of an electron in an artificial atom
AdvantagesLong decoherence time
Future Scalabilty
Artifical atoms are bigger than regular atoms therefore easier to manipulate
Decoherence time ~ 100ns
bull Time before the quantum mechanical system starts acting in a classical way with its complex environment
bull The state of the system has not yet collapsed due to (unwanted) environmental effects
bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error
correction This requirement is met
Artificial Atombull Double Barrier
Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength
rarr Discrete energy levels
Adding Electrons changing Vgate
bull 2D-Harmonic Oscillator
bull Shell structure as in atoms
bull Magic Numbers 2 6 12
bull To add ldquoevenrdquo electron requires only additional Coulomb energy
Comparison with Hydrogenbull Artificial Atom
Energy levels ~ 1meV
Size ~ 10μm
Weak magnetic fields can affect energy levels
bull Hydrogen
Energy levels ~ 1eV
Size ~ 1Aring
Only strong magnetic fields can perturb energy levels
Factor 1000
Tuning the Quantum Dot
bull Tune so we have one valence electron
bull Initial state can be set by applying homogeneous magnetic field rarr |0gt
bull Low temperature kT lt ΔE (state gap)
bull Now we have defined our single qubit
Energy
position
Gate bias
Spin up - electron
Unoccupied state
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Molecular Beam Epitaxy
bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to
combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms
only meet each other on the wafer
Spin Quantum Computing
Qubit information is stored in the spin state of an electron in an artificial atom
AdvantagesLong decoherence time
Future Scalabilty
Artifical atoms are bigger than regular atoms therefore easier to manipulate
Decoherence time ~ 100ns
bull Time before the quantum mechanical system starts acting in a classical way with its complex environment
bull The state of the system has not yet collapsed due to (unwanted) environmental effects
bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error
correction This requirement is met
Artificial Atombull Double Barrier
Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength
rarr Discrete energy levels
Adding Electrons changing Vgate
bull 2D-Harmonic Oscillator
bull Shell structure as in atoms
bull Magic Numbers 2 6 12
bull To add ldquoevenrdquo electron requires only additional Coulomb energy
Comparison with Hydrogenbull Artificial Atom
Energy levels ~ 1meV
Size ~ 10μm
Weak magnetic fields can affect energy levels
bull Hydrogen
Energy levels ~ 1eV
Size ~ 1Aring
Only strong magnetic fields can perturb energy levels
Factor 1000
Tuning the Quantum Dot
bull Tune so we have one valence electron
bull Initial state can be set by applying homogeneous magnetic field rarr |0gt
bull Low temperature kT lt ΔE (state gap)
bull Now we have defined our single qubit
Energy
position
Gate bias
Spin up - electron
Unoccupied state
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Spin Quantum Computing
Qubit information is stored in the spin state of an electron in an artificial atom
AdvantagesLong decoherence time
Future Scalabilty
Artifical atoms are bigger than regular atoms therefore easier to manipulate
Decoherence time ~ 100ns
bull Time before the quantum mechanical system starts acting in a classical way with its complex environment
bull The state of the system has not yet collapsed due to (unwanted) environmental effects
bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error
correction This requirement is met
Artificial Atombull Double Barrier
Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength
rarr Discrete energy levels
Adding Electrons changing Vgate
bull 2D-Harmonic Oscillator
bull Shell structure as in atoms
bull Magic Numbers 2 6 12
bull To add ldquoevenrdquo electron requires only additional Coulomb energy
Comparison with Hydrogenbull Artificial Atom
Energy levels ~ 1meV
Size ~ 10μm
Weak magnetic fields can affect energy levels
bull Hydrogen
Energy levels ~ 1eV
Size ~ 1Aring
Only strong magnetic fields can perturb energy levels
Factor 1000
Tuning the Quantum Dot
bull Tune so we have one valence electron
bull Initial state can be set by applying homogeneous magnetic field rarr |0gt
bull Low temperature kT lt ΔE (state gap)
bull Now we have defined our single qubit
Energy
position
Gate bias
Spin up - electron
Unoccupied state
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Decoherence time ~ 100ns
bull Time before the quantum mechanical system starts acting in a classical way with its complex environment
bull The state of the system has not yet collapsed due to (unwanted) environmental effects
bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error
correction This requirement is met
Artificial Atombull Double Barrier
Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength
rarr Discrete energy levels
Adding Electrons changing Vgate
bull 2D-Harmonic Oscillator
bull Shell structure as in atoms
bull Magic Numbers 2 6 12
bull To add ldquoevenrdquo electron requires only additional Coulomb energy
Comparison with Hydrogenbull Artificial Atom
Energy levels ~ 1meV
Size ~ 10μm
Weak magnetic fields can affect energy levels
bull Hydrogen
Energy levels ~ 1eV
Size ~ 1Aring
Only strong magnetic fields can perturb energy levels
Factor 1000
Tuning the Quantum Dot
bull Tune so we have one valence electron
bull Initial state can be set by applying homogeneous magnetic field rarr |0gt
bull Low temperature kT lt ΔE (state gap)
bull Now we have defined our single qubit
Energy
position
Gate bias
Spin up - electron
Unoccupied state
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Artificial Atombull Double Barrier
Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength
rarr Discrete energy levels
Adding Electrons changing Vgate
bull 2D-Harmonic Oscillator
bull Shell structure as in atoms
bull Magic Numbers 2 6 12
bull To add ldquoevenrdquo electron requires only additional Coulomb energy
Comparison with Hydrogenbull Artificial Atom
Energy levels ~ 1meV
Size ~ 10μm
Weak magnetic fields can affect energy levels
bull Hydrogen
Energy levels ~ 1eV
Size ~ 1Aring
Only strong magnetic fields can perturb energy levels
Factor 1000
Tuning the Quantum Dot
bull Tune so we have one valence electron
bull Initial state can be set by applying homogeneous magnetic field rarr |0gt
bull Low temperature kT lt ΔE (state gap)
bull Now we have defined our single qubit
Energy
position
Gate bias
Spin up - electron
Unoccupied state
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Adding Electrons changing Vgate
bull 2D-Harmonic Oscillator
bull Shell structure as in atoms
bull Magic Numbers 2 6 12
bull To add ldquoevenrdquo electron requires only additional Coulomb energy
Comparison with Hydrogenbull Artificial Atom
Energy levels ~ 1meV
Size ~ 10μm
Weak magnetic fields can affect energy levels
bull Hydrogen
Energy levels ~ 1eV
Size ~ 1Aring
Only strong magnetic fields can perturb energy levels
Factor 1000
Tuning the Quantum Dot
bull Tune so we have one valence electron
bull Initial state can be set by applying homogeneous magnetic field rarr |0gt
bull Low temperature kT lt ΔE (state gap)
bull Now we have defined our single qubit
Energy
position
Gate bias
Spin up - electron
Unoccupied state
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Comparison with Hydrogenbull Artificial Atom
Energy levels ~ 1meV
Size ~ 10μm
Weak magnetic fields can affect energy levels
bull Hydrogen
Energy levels ~ 1eV
Size ~ 1Aring
Only strong magnetic fields can perturb energy levels
Factor 1000
Tuning the Quantum Dot
bull Tune so we have one valence electron
bull Initial state can be set by applying homogeneous magnetic field rarr |0gt
bull Low temperature kT lt ΔE (state gap)
bull Now we have defined our single qubit
Energy
position
Gate bias
Spin up - electron
Unoccupied state
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Tuning the Quantum Dot
bull Tune so we have one valence electron
bull Initial state can be set by applying homogeneous magnetic field rarr |0gt
bull Low temperature kT lt ΔE (state gap)
bull Now we have defined our single qubit
Energy
position
Gate bias
Spin up - electron
Unoccupied state
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Single Qubit manipulation
bull Unitary operations can be made by applying a local magnetic field H
ZE = -μB = g
μB SB
bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc
(Magnetic force microscope tip)
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Two Qubit Manipulation
bull Complete set of logic requires a CNOT
bull Dots are placed so close that they overlap and interact
bull Hspin
= J(t)S1S
2
Exchange couplingJ(tEB) = E
triplet -E
singlet
(4th order Harmonic Oscillator)
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Ground State Splitting (J = Et ndash E
s)
bull 2 coupled fermions must have an total anti-symmetric wave function
bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ
sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)
bull The triplet states are (|darrdarrgt)|ψ
tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)
(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Solving J(B(t)) Exchange Coupling
bull Different solutions Heitler-London Hund-Mulliken Hubbard
bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
SWAP - gatebull Assume J can be pulsed
J(t) = 0 J0
Formula 1
Formula 2
bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
XOR ~ CNOT
bull Formula 3
bull Requirements Spin rotations about the z-axis Squareroot of U
swap
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Read out Memory
bull Assume dot with an electron with some information stored in spin-state
bull Connect two leads to dot
bull Apply a small bias (ΔV) rarr Current (i)Energy
position
Gate bias
Spin up - electron
Unoccupied state
i
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Another Spin up electron enters dot
bull Pauli principle forces electrons with spin up to occupy the higher energy state
bull Negligible chance of tunneling
E
position
Gate bias
Spin up - electron
Higher energy level(forbidden classically)
i=0
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Spin down electron enters dot
bull Pauli principle allows the new electron to join the same energy level as the original electron
bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow
E
position
Gate bias
Spin up - electron
Unoccupied state
ine0
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Read out Memory
bull We have a way of measuring the spin state of an electron in a quantum dot
bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties
bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique
bull The electron current can be specialized (we can aim its spin to make measurement efficient)
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
The Physical System Excitons Trapped in GaAs Quantum Dots
bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid
bull Often formed when photons excite electrons from the valence band into the conduction band
bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses
bull Decay by radiating photons Decay time ~50ps-1ns
bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Initialization
bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the
register
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Relatively Long Decoherence Times
bull Mechanismsndash Radiative Decay ~10ps-1ns
bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations
bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral
bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
A Universal set of Quantum Gates
bull Single Qubit Rotations through laser induced Rabi Oscillations
bull CNOT operations through dipole interactions and laser excitation
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Single Qubit Gates Rabi Flopping
bull Light-particle interaction is characterized by the product of the dipole moment and the electric field
μbullE(t)= ħR(t)
Where R(t) is the Rabi frequency and the pulse area is given by
Θ(t)=intR(t)dt
and the state at time t is then given by
Cos(Θ2)|0gt+Sin(Θ2)|1gt
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Stufler et al
Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses
Cos(Θ2)|0gt+Sin(Θ2)|1gt
|1gt =gt electric charge
=gtPhotocurrent (PC)PC~Sin2(Θ2)
π-pulse corresponds to a population inversion
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
CNOT Dipole Coupling
Nearest neighbor interactions alter the energy states
Effective energy Ersquoi = Ei + sumjnei ∆Eij nj
Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Overcoming Short Interaction Distances
bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs
bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots
bull Optical Cavity coupling and fiber optical interconnects have also been proposed
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
Read Out of Specified Qubit States
bull Optical readoutExcitons decay spontaneously and the resulting radiation can be
measured
Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD
Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination
In either case repeated measurements have to be made A single shot readout still needs to be developed
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
5 DiVincenzo QC Criteria
1 A scalable physical system with well-characterized qubits
2 The ability to initialize the state of the qubits to a simple fiducial state
3 Relatively long decoherence times compared to gate-operation times
4 A universal set of quantum gates
5 Qubit-specific measurement capability
- Quantum Dots
- Implementing Quantum Computers
- What is a quantum dot
- Gallium Arsenide Quantum Dots
- Energy Band Levels
- Slide 6
- Slide 7
- Tuning Quantum Dots
- Manufacturing methods
- Electron Beam Lithography
- Slide 11
- Molecular Beam Epitaxy
- Spin Quantum Computing
- Decoherence time ~ 100ns
- Artificial Atom
- Adding Electrons changing Vgate
- Comparison with Hydrogen
- Tuning the Quantum Dot
- Single Qubit manipulation
- Two Qubit Manipulation
- Ground State Splitting (J = Et ndash Es)
- Solving J(B(t)) Exchange Coupling
- SWAP - gate
- XOR ~ CNOT
- Read out Memory
- Another Spin up electron enters dot
- Spin down electron enters dot
- Slide 28
- 5 DiVincenzo QC Criteria
- The Physical System Excitons Trapped in GaAs Quantum Dots
- Initialization
- Relatively Long Decoherence Times
- A Universal set of Quantum Gates
- Single Qubit Gates Rabi Flopping
- Stufler et al
- CNOT Dipole Coupling
- Overcoming Short Interaction Distances
- Read Out of Specified Qubit States
- Slide 39
-
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