quantum information processing with semiconductors
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Quantum Information Processing with Semiconductors. Martin Eberl, TU Munich JASS 2008, St. Petersburg. Overview. Quantum Computation Quantum bits Quantum gates Quantum parallelism Deutsch - Algorithm Semiconductor quantum computer Self-assembled quantum dots - PowerPoint PPT PresentationTRANSCRIPT
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Quantum Information Processing with Semiconductors
Martin Eberl, TU MunichJASS 2008, St. Petersburg
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Overview Quantum Computation
Quantum bits Quantum gates Quantum parallelism Deutsch - Algorithm
Semiconductor quantum computer Self-assembled quantum dots SRT with SiGe heterostructures Donor-based quantum computing Quantum bits Hyperfine structure Quantum gates Readout Calibration
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Quantum bit (qubit)
classical bit:0 or 1 ⇔
qubit:0 or 1 or superposition
measurement: either with probability
or with probability
(normalization)
After measurement: Collapse of the wave function
or
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Quantum gates= logical operation on qubits
classical: quantum:
Representation of quantum gates:Unitary matrices:
NOT- gate
Single-qubit gate: NOT- gate
(adjoint = transpose & complex conjugate)
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Hadamard gate
pure state → mixed state
Only 1 classical single-bit gate, but ∞ single-qubit gates
H² = 1
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Two qubits
Probability for measuring first qubit 0:
After measuring 1st qubit 0:
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Two-qubit states• product state:
for example
⇒ Measurement of 1st qubit doesn‘t affect the 2nd one• entangled state:
not writeable as a product state
Bell state:
Measurement of 1st qubit = 0 (with probability 0.5)then 2nd qubit must be 0 too
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Two-qubit gates Iclassical: AND, NAND, OR, NOR, XOR, XNOR
⇒ NAND is universal2 bits input → 1 bit output ⇒ not reversible
quantum: CNOTcontrol target
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Two-qubit gates II
is unitary ⇒ reversible (bijection)
CNOT is universal:every logical operation can be performed by CNOT + single-qubit gates
Operation on state:
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No-Cloning-Theoremit‘s impossible to copy arbitrary quantum states
proof:
only true for 0 or 1only pure states can be copied
copy with CNOTdata space \ /
CNOT
CNOT
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Function evaluationunitary transformation Uf:
Uf
By carrying along, it is possible to use a non bijective function as a unitary one
picture of a controlled operation
ffor f(x) = x we get CNOT
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Quantum parallelism Iquantum register of n qubits:create mixed state:
for n = 3:
== =
Superposition of 2n states
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Quantum parallelism II
Uf
H
H
H
……
entangled state
for n = 3:⇒ simultaneous evaluation of f(x) for 2n arguments! problem: measurement gives random f(x)
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Deutsch – Algorithm I4 possible functions
{constantfunctions
{balancedfunctions
Problem: determinate if a function f(x) isbalanced or constant
Classical: 2 function calls needed
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Deutsch – Algorithm II
create superposition:
Uf
H
H
H
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Deutsch – Algorithm III
evaluate f (note that and ) _
___ ___
___ ___
→Uf
___ ___
___ ___UH
UH{ constant
balanced|
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Advantages
Only for certain problems: exploitation of special properties:
e.g. period, correlation Deutsch-Algorithm⇒ ⇒ Shor‘s Algorithm (prime-factoring)
Repetition of the same task on large number of input values
e.g. search through an unstructured database (Grover‘s Algorithm)
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Self-assembled quantum dots
• quantum dots self-assembled by growing InAs over GaAs • Excitons (electron-hole pairs) used as qubits ⇒ created by light absorption ⇒ confined in quantum dots• 4-8 nm distance ⇒ overlap of wave functions ⇒ tunneling
Dot 1 Dot 2 Dot 1 Dot 2 Dot 1 Dot 2 Dot 1 Dot 2
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Spin resonance transistor with SiGe heterostructures
• heterostructure of different SixGe1-x layers ⇒ Landé g-factor changes• spin of weakly bound electron from 31P represents the qubit• Voltage at gate pulls wave function away from donor• different g-factor ⇒ resonance frequency changes• magnetic field in resonance performs logical operations
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Donor-based quantum computing
Design:Brf ≅ 10-3 Tesla
B ≅ 2 Tesla
T 100 mK≅
A J A
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Overview
Only Si – Isotopes with nuclear spin In = 0 31P – Donors have In = ½ Nuclear spin of donors is used for qubits Logical operations are performed with different
voltages on the gates above the donors in combination with the magnetic field Brf
Initialization and measurement is made by gauging electron charges
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Nuclear spin as qubitProblem in general:
Interaction of quantum system with environment ⇒ decay of information (decoherence time) ⇒ computation must be completed before the
information has significantly decayed
Solution: nuclear spinlittle interaction large decoherence time⇒(estimated to be in the order of 1018 s at mK temperatures)
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Electron structure
Low temperature T 100 mK≅ ⇒ no electrons in the conduction band ⇒ isolator
Phosphorus is a group V element ⇒ one additional electron, which is very
weakly bound, close to the conduction band ⇒ Similar to a Hydrogen atom with bigger
radius and smaller energy
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Hyperfine structure I
electron nucleus interaction
Probability density of electron wave function at nucleus
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} Δf
Hyperfine structure II
Logical operations between electron and nucleus:SWAP-Operation:
⇒ Transfer of nuclear spin state to electronCNOT:
= frequency for Brf
to perform SWAP
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Single-qubit gates I
Precession of nuclear spin around B with the Larmor frequency B
spinBring Brf into resonance with spin precession
⇒ arbitrary rotation possible
Problem: Brf is globally applied, not locally
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Single-qubit gates II
Lab frame Rotation frame
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Single-qubit gates IIILarmor frequency is dependent on the hyperfine interaction of the electron with the nucleus
Apply voltage at the A-Gate: ⇒ electron is drawn away from the
nucleus ⇒ Larmor frequency for single donor
changes ⇒ it’s possible to address nuclear spin
of single donor with Brf
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Two-qubit gatesApply positive electric field on J-Gate turn electron ⇒mediated interaction between nuclei on or off
New hyperfine structure for the system of both nuclei and their electrons
Magnetic field Brf can modify the spin states of the system and thus perform logical operations like SWAP or CNOT
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Readout
Qubit stored in nucleus spin ⇒ little interaction with the environment ⇒ hard to read out
SWAP between nucleus and electronImportant: fast read out, before information decays
Spin measurement possible, but too slow ⇒ charge measurement
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Readout Prepare electron spin of 1st donor in a known state Transfer electron from 2nd donor using A-Gate
voltage only possible, if spin is pointing in different ⇒
direction Perform charge measurement
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CalibrationVariation of donor positions and gate sizes it’s necessary to calibrate each gate⇒
• set Brf = 0 and measure nuclear spin• switch Brf on and sweep through small voltage interval at A-Gate• measure nuclear spin again
it will only flip, if resonance occurred in the A-⇒Gate voltage range• After A-Gates have been calibrated, use same procedure with the J-Gates• Calibration can be performed parallel on many Gates, resonance voltages can be stored on capacitors
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Challenges for building the computer Silicon completely free of spin &
charge impurities Donors in an ordered array ~ 25 nm
beneath the surface Very small gates must be placed on
the surface right above the donorsAdvantage to other quantum computer concepts: it’s possible to incorporate 106 qubits
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Quantum Information Processing with Semiconductors
Nielsen, Chuan, Quantum computation and quantum information, 2001 Stolze, Suter, Quantum computing, 2004 Chen et. al., Optically induced entanglement of excitons in a single quantum dot, 2000 Rutger Vrijen et. al., Electron spin resonance transistors for quantum computing in silicon-germanium heterostructures, 2000 B.E. Kane, A silicon-based nuclear spin quantum computer, Nature 393: 133-137, 1998. B.E. Kane, Silicon-based quantum computation, 2008