slides chap2
TRANSCRIPT
Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Qubit: The Unit of Quantum Computation
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Lecture 1
Qubit, Pauli matrices, 2D operators
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Qubit: The Unit of Quantum Computation1 Qubit: Definition & Properties
What is a qubit?Bloch Sphere Representation
2 Pauli Matrices & 2D OperatorsIntroductionPropertiesOperators in 2D Hilbert Space
3 Quantum GatesGatesExamples√NOT Gate
Decomposition of Single-qubit Gates4 Composite systems
Product StatesMultiple QubitsControlled Gates
5 Physical Realization 3 / 23
Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Qubit
The bit (binary digit) is the basic unit of classicalcomputation, having two values – 1 (on) and 0 (off); atwo-state device, e.g., a transistor
The quantum bit is the unit of a quantum computer. It is atwo-level quantum system, represented by a vector in 2DHilbert Space, whose basis states can be labelled as
|0〉 =
(10
), |1〉 =
(01
)analogous to the on and off states of a classical bit. Physicalexamples of qubit: an electron spin (along a given direction)with two eigenstates | ↑〉 and | ↓〉, or vertical and horizontalpolarization states of a photon, or the ground and excitedstates of an atom.
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Qubit...
The basis {|0〉, |1〉} is known as the standard basis or thecomputational basis in 2D Hilbert space.
The most general state of a qubit is |ψ〉 = α|0〉+ β|1〉. Thequbit can exist in a superposition of on and off states, unlikea classical bit. This property is of paramount importance toquantum computing.
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Bloch Sphere Representation
For a normalized qubit, |α|2 + |β|2 = 1. This suggests arepresentation in spherical polar coordinates,
|ψ〉 = e−iφ
2 cos(θ
2)|0〉+ e
iφ2 cos(
θ
2)|1〉, (0 ≤ θ ≤ π; 0 ≤ φ ≤ 2π)
This is called the Bloch Sphere Representation
qubit is thus a point onthe surface of a sphere ofunit radius, parametrizedby the coordinates (θ, φ).
The basis states |0〉 and|1〉 are then represented bythe north and south polesof the sphere, respectively.
Figure : Qubit on Bloch sphere.
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Pauli Matrices
Any quantum system in 2D Hilbert Space, e.g., an electron spin,can be adequately described by the 3 Pauli matrices:
σx =
(0 11 0
), σy =
(0 −ii 0
), σz =
(1 00 −1
)
The corresponding electron spin operators areSx = ~
2σx , Sy = ~2σy , Sz = ~
2σz .
The computational basis states, |0〉 and |1〉 are theeigenstates of σz , corresponding to spin-up and spin-downstates along z-direction.
The eigenstates of σx are, |+〉 = 1√2
(|0〉+ |1〉) and
|−〉 = 1√2
(|0〉 − |1〉).
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Pauli Matrices: Properties
1 σx , σy , σz are Hermitian and Unitary, i.e., σ†i σi = σ2i = I.
2 σx , σy , σz have two eigenvalues, ±1.
3 det(σi ) = 1.
4 Trace(σi) = 0.
5 σiσj = iεijkσk ∀ i 6= j .
6 From (5), it follows that [σi , σj ] = 2iεijkσk if i 6= j .
7 From (1) and (5), the anti-commutator{σi , σj} = σiσj + σjσi = δij .
8 Also, (1) and (5) can be condensed into σiσj = δij I + εijkσk9 From (1) and (6), for a unit vector n̂,
(n̂ · σ)2 = (nx · σx + ny · σy + nz · σz)2 = I
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
2D Operators
The three σi and I2 form an independent set.
Any 2× 2 unitary matrix has four independent parameters.
The Pauli matrices, along with I2 form a complete basis in thespace of 2× 2 unitary matrices.
Therefore, any operator in 2D Hilbert Space can be written downas:
U =3∑
i=1
ciσi + c4I
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Rotation Operator
The operator U[R(θ)] = e−i~θ·σ/2 causes rotations in 2D Hilbert
Space. On expanding the exponential and grouping terms, we find:
e−i~θ·σ/2 = cos
(θ
2
)I− i sin
(θ
2
)(θ̂ · σ)
This will prove very useful later on for constructing variousquantum gates.
Example
Take the operator for rotation along x-axis, U[Rx(θ)] = e−iθσx/2
e−iθσx/2 =
(cos(θ/2) −i sin(θ/2)−i sin(θ/2) cos(θ/2)
)Using the Bloch sphere, examine its action on the |0〉, |1〉 states.
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Lecture 2
Quantum gates, decomposition of single-qubit gates
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Quantum Gates
In classical computation, logic operations on bitsare represented by gates, e.g., the NOT gateflips a bit:
Input Output
0 11 0
Quantum gates are quite clearly unitary operators.
As we have already seen, all single qubit gates can berepresented in terms of Pauli matrices and the identity matrix.
The quantum analogue of the NOT gate is σx .
σx |0〉 =
(0 11 0
)(10
)=
(01
)= |1〉; σx |1〉 = |0〉
Since for every U there is a U†, all quantum gates arereversible.
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Quantum Gates...
Quantum mechanics allows construction of gates that have noclassical analogue. Two important single qubit gates are:
the phase gate,
S =
(1 00 i
); S
(|0〉+ |1〉√
2
)=|0〉+ i |1〉√
2
which introduces a relative phase of π,
the Hadamard gate,
H =1√2
(1 11 −1
); H|0〉 =
|0〉+ |1〉√2
, H|1〉 =|0〉 − |1〉√
2
which transforms the computational basis to the |±〉 statesand vice versa
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
√NOT Gate
Another example of a uniquely quantum gate is the√NOT gate.
To construct this, we note that:
1
√NOT
√NOT = σx
2 σx =
(0 11 0
)= iU[Rx(π)]
Thus, up to an overall phase, which is of no importance whilecalculating probabilities,
√NOT = U[Rx(π/2)] =
1√2
(1 −i−i 1
)
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Decomposition of Single-qubit Gates
In the derivation of√NOT gate, we expressed the NOT gate
as a rotation operator. More generally, any single qubit unitaryoperation can be written as U = e iαRn̂(θ), where n̂ is the axisand θ is the angle of rotation, and e iα is a global phase shift.
Any single qubit gate can be decomposed into variouscombinations of rotations and global phase shifts. Particularlyimportant is the Z-Y decomposition:
U = e iαRz(β)Ry (γ)Rz(δ)
Analogous decompositions can be carried out in terms ofrotations along any two non-parallel axis n̂ and m̂:U = e iαRn̂(β)Rm̂(γ)Rn̂(δ)
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Decomposition of Single-qubit Gates...
Using the Z-Y decomposition, it can be shown that for anyunitary U we can find unitary A, B, C such that ABC = Iand U = e iαAσxBσxC
Example (Decomposition of Hadamard)
H = Rz(3π)Ry (π
2)Rz(π) (Verify)
1√2
(1 11 −1
)=
(i 00 −i
)(cos(π/8) − sin(π/8)sin(π/8) cos(π/8)
)σx (1)(
cos(π/8) sin(π/8)− sin(π/8) cos(π/8)
)(−1 00 −1
)σx
(i 00 −i
)
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Lecture 3
Composite systems, product representation, controlled gates,physical realization
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Product States
For computational purposes we require more than one qubit. Tostudy multi-qubit states, the tensor-product notation is useful.
Suppose you have two systems A and B in states |ψ1〉A and|ψ2〉b respectively. The state of the composite system A+Bcan be written in the form |ψ1〉A⊗ |ψ2〉B where ⊗ denotes thetensor product.
In matrix notation, product states are expressed as follows. If
〈ψ|1 =(a1 a2 a3 · · · an
)〈φ|2 =
(b1 b2 b3 · · · bn
)Then
〈ψφ| = 〈ψ|1 ⊗ 〈φ|2=
(a1b1 a1b2 · · · a1bn a2b1 a2b2 · · · a2bn
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Multiple Qubits
States with n qubits can have states of the form |000...00〉,|111...11〉, |010...00〉 etc. and their superpositions. The states
1√n⊗n
i=1,xi∈{0,1}|xi 〉
form the computational basis in a n-qubit system.
Example (2-qubit system)
The states |00〉, |01〉, |10〉, |11〉 form the computational basis fora 2-qubit system.
|00〉 =
(10
)⊗(
10
)=
1000
, |01〉 =
(10
)⊗(
01
)=
0100
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
The CNOT gate
The CNOT or controlled-NOT gate is a 2-qubit gate that flipsthe 2nd qubit only if the first qubit is in the |1〉 state.
Action of CNOT on 2-qubit computational basis
|00〉 CNOT−−−−→ |00〉, |01〉 CNOT−−−−→ |01〉
|10〉 CNOT−−−−→ |11〉, |11〉 CNOT−−−−→ |10〉
Equivalently, |A,B〉 CNOT−−−−→ |A,A⊕ B〉 where ⊕ denotes additionmodulo 2.
The matrix representation ofCNOT is
1 0 0 00 1 0 00 0 0 10 0 1 0
The circuit diagram is
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Controlled Gates
The CNOT is an example of a broad class of controlled gates thatuse a single qubit as control and act only if the control is set to |1〉.
A controlled- U gate is represented in
matrix form as
(I2 00 U
)where U
can be an operator of any dimension.
Figure : Controlled-U
Note
Any arbitrary gate can be constructed using only single-qubitoperations and CNOT.
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
Physical Realization of Qubits
As mentioned before, any two-level quantum system can be aqubit.
In NMR quantum computers, nuclear spins are used as qubitswith spin-up and spin-down states as the computational basis.Gates are applied by magnetic pulses on these nuclear spins.
In superconducting phase qubit, the direction (phase) ofcurrent (clockwise or anticlockwise) is the quantum state.
In optical quantum computers, the |0〉 and |1〉 states aredefined as the states of photons being in either of twopotential wells.
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Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization
References
M.A. Nielsen & I.L. Chuang: Quantum Computation andQuantum Information, Cambridge University Press, ISBN9781107002173
John Preskill, Lecture Notes on Quantum Computation,http://www.theory.caltech.edu/~preskill/ph219/
David Deutsch, Lectures on Quantum Computation
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