1 Quantum Logic Gates and Circuits • Quantum Gates: Building Blocks of Quantum Computers • Quantum Gate transforms a Quantum State to a New State • State Transformations Performed by Gates Described by Hermitian Operators • Matrix Describing State Transformation is Transfer Matrix Quantum Logic Gate Matrices • Matrices are Unitary • Transformations are Reversible • Unitary Transformations Corresponds to: – Length Preservation – Information Preserving Rotation in Vector Space U −1 = U † UU † = U † U = I
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1
Quantum Logic Gates and Circuits
• Quantum Gates: Building Blocks of Quantum Computers
• Quantum Gate transforms a Quantum State to a New State
• State Transformations Performed by Gates Described by Hermitian Operators
• Matrix Describing State Transformation is Transfer Matrix
Quantum Logic Gate Matrices• Matrices are Unitary
• Transformations are Reversible • Unitary Transformations
Corresponds to: – Length Preservation – Information Preserving Rotation in Vector
Space
U−1 = U†
UU† = U†U = I
2
Quantum Logic Gate Matrices• Gates have equal number of Inputs
• Classical Logic – Typically is Irreversible Logic – Reversible Logic is a special case – Fanout is a powerful use-case
• Quantum Logic – Comparison with Classical – No Cloning theorem – Universal Quantum Logic Gates
3
Classical Irreversible Logic
• Theory of Nature of Computing (Church, Turing 1936)
• Universality of Primitive Operationsx y x AND y
0 0 0
0 1 0
1 0 0
1 1 1
x y x OR y
0 0 0
0 1 1
1 0 1
1 1 1
y NOT y
0 11 0
Types of Reversibility
• Logical Reversibility – Ability to reconstruct input from output. Circuit
function is a Bijection. • Bijection implies two properties: (1) one-to-one, (2) onto
• Physical Reversibility – Thermodynamic entropy based arguments that
relate the loss of information to an increase in dissipated heat.
– Heat dissipation during a computation is generally a sign of physical irreversibility.
4
Thermodynamics Concepts (Oversimplified)
• Thermodynamics – branch of physics that studies the effects of changes in
temperature, pressure, and volume in physical systems
• Physical System – A region of spacetime and all entities (particles and
fields) contained within it. (eg. universe, transistors, circuits, computers - defn from M. Frank)
• Entropy – measure of the amount of energy in a physical system
that cannot be used to do work - entropy S is multiplied by a temperature to yield an amount of energy. It is a measure of the disorder and randomness present in a system. A quantitative measure of the amount of thermal energy NOT AVAILABLE to do work.
Physical Irreversibility• 2-input NAND Gate
– one output, two inputs – in computing an output, one input is “erased” – information irretrievably lost – change in entropy of the system is one bit of
information - quantitatively this is ln 2 – conversion to energy increase of kT ln 2 where k
is Boltzman’s constant and T is temperature – corresponds to energy “lost” to heat dissipation
and a sign of physical irreversibility
5
Developments in Reversibility• Can a computation be accomplished in a
logically reversible fashion? (unlike using a NAND gate - 1970’s)
• Must heat be dissipated during a computation? – Feynmann points out (1986) transistor
dissipates 1010kT joules of heat, DNA copying in a human cell dissipates 100 kT joules all far from 0.693 kT joule lower bound from erasing a single bit
Trend of minimum transistor switching energy
1
10
100
1000
10000
100000
1000000
1995 2005 2015 2025 2035
Year of First Product Shipment
Min
tra
nsi
sto
r sw
itch
ing
en
erg
y, k
Ts
High
Low
trend
½CV2 based on ITRS ‘99 figures for Vdd and minimum transistor gate capacitance. T=300 K *based on chart prepared by M. Frank at Univ. of Fla.
Min. Transistor Switching Energy Trend*
6
Developments in Reversibility• 1973 - Bennett proved that classical
computation can be accomplished with no energy dissipated per computational step and with reversibility (reversible Turing machine model)
• This triggered a search for physical models for reversible classical computation
• Common Model is a discrete one-to-one binary-valued Boolean function with an equal number of inputs and outputs
Reversible Logic Circuit
• f is a bijective function • contains symmetry that allows for other
forms of representation (transformation matrix)
x1x2
xn
y1y2
yn
Reversible Function �
f
7
Classical Reversible Gates/Operators
NOT
In Out 0 1 1 0
0 11 0⎡
⎣⎢
⎤
⎦⎥
Classic Symbol Physically Irreversible Logically Reversible
Classic Truth Table Notion of Inputs/Outputs
Symbol for Reversible NOT Gate
(aka Pauli-X)
Matrix Representation of Pauli-X Gate
Functionality
Reversible NOT Gate
NOT
0 11 0⎡
⎣⎢
⎤
⎦⎥
Symbol for Reversible NOT Gate
Matrix Representation of NOT Gate Functionality
0 11 0⎡
⎣⎢
⎤
⎦⎥
10⎡
⎣⎢⎤
⎦⎥ =
01⎡
⎣⎢⎤
⎦⎥
0 11 0⎡
⎣⎢
⎤
⎦⎥
01⎡
⎣⎢⎤
⎦⎥ =
10⎡
⎣⎢⎤
⎦⎥
| 0〉 |1〉 | 0〉 |1〉
8
Reversible NOT Gate
NOT A =
0 11 0⎡
⎣⎢
⎤
⎦⎥
Symbol for Reversible NOT Gate
Matrix Representation of NOT Gate Functionality
A† =
0 11 0⎡
⎣⎢
⎤
⎦⎥
A†A =
0 11 0⎡
⎣⎢
⎤
⎦⎥
0 11 0⎡
⎣⎢
⎤
⎦⎥ =
1 00 1⎡
⎣⎢
⎤
⎦⎥ = I
Reversible NOT Gate
NOT
0 11 0
⎡
⎣⎢
⎤
⎦⎥ = 0 1 + 1 0
Symbol for Reversible NOT Gate
Matrix Representation of NOT Gate Functionality
0 11 0⎡
⎣⎢
⎤
⎦⎥
10⎡
⎣⎢⎤
⎦⎥ =
01⎡
⎣⎢⎤
⎦⎥
Dirac Notation Example
0 1 + 1 0( ) 0
= 0 1 0 + 1 0 0
= 0 0( ) + 1 1( ) = 1
9
Reversible NOT Gate
NOT
0 11 0
⎡
⎣⎢
⎤
⎦⎥ = 0 1 + 1 0
Symbol for Reversible NOT Gate
Matrix Representation of NOT Gate Functionality
0
1
0 1
0 11 0
⎡
⎣⎢⎢
⎤
⎦⎥⎥
“INPUTS”
“OUTPUTS”
Permutation Matrix Transformations of
Qubits
Derivation of I2 or σ0
• This Operator Performs an Identity Transformation of the Basis Vectors:
• Computed as: | 0〉!| 0〉 |1〉!|1〉
σ 0 = I =| 0〉〈0 | + |1〉〈1|
σ 0 = I = 1
0⎡⎣⎢⎤⎦⎥⊗ 1 0⎡⎣ ⎤⎦ +
01⎡⎣⎢⎤⎦⎥⊗ 0 1⎡⎣ ⎤⎦
σ 0 = I = 1 0
0 0⎡⎣⎢
⎤⎦⎥+ 0 0
0 1⎡⎣⎢
⎤⎦⎥= 1 0
0 1⎡⎣⎢
⎤⎦⎥
10
Derivation of X or σX
• This Operator “Flips” or “Negates” a Qubit:
• Computed as: | 0〉!|1〉 |1〉!| 0〉
σ1 = X =| 0〉〈1| + |1〉〈0 |
σ1 = X = 1
0⎡⎣⎢⎤⎦⎥⊗ 0 1⎡⎣ ⎤⎦ +
01⎡⎣⎢⎤⎦⎥⊗ 1 0⎡⎣ ⎤⎦
σ1 = X = 0 1
0 0⎡⎣⎢
⎤⎦⎥+ 0 0
1 0⎡⎣⎢
⎤⎦⎥= 0 1
1 0⎡⎣⎢
⎤⎦⎥
Derivation of Y or σY
• This Operator Multiplies a Qubit by i then “Flips” or “Negates” it:
• Computed as: | 0〉! i |1〉 |1〉! −i | 0〉
σ 2 = Y = −i | 0〉〈1| +i |1〉〈0 |
σ 2 = Y = −i 1
0⎡⎣⎢⎤⎦⎥⊗ 0 1⎡⎣ ⎤⎦ + i 0
1⎡⎣⎢⎤⎦⎥⊗ 1 0⎡⎣ ⎤⎦
σ 2 = Y = −i 0 1
0 0⎡⎣⎢
⎤⎦⎥+ i 0 0
1 0⎡⎣⎢
⎤⎦⎥= 0 −i
i 0⎡⎣⎢
⎤⎦⎥
11
Derivation of Z or σZ
• This Operator is an Identity with a 180 Degree Phase Shift Operation:
• Computed as: | 0〉!| 0〉 |1〉! − |1〉
σ 3 = Z =| 0〉〈0 | − |1〉〈1|
σ 3 = Z = 1
0⎡⎣⎢⎤⎦⎥⊗ 1 0⎡⎣ ⎤⎦ −
01⎡⎣⎢⎤⎦⎥⊗ 0 1⎡⎣ ⎤⎦
σ 3 = Z = 1 0
0 0⎡⎣⎢
⎤⎦⎥− 0 0
0 1⎡⎣⎢
⎤⎦⎥= 1 0
0 −1⎡⎣⎢
⎤⎦⎥
Pauli Operator Examples• Assume the Following:
|ϕ〉 = σ i |ψ 〉 = σ i[α0 | 0〉 +α1 |1〉]
|ϕ〉 = σ 0 |ψ 〉 = 1 0
0 1⎡⎣⎢
⎤⎦⎥α0
α1
⎡
⎣⎢
⎤
⎦⎥ =
α0
α1
⎡
⎣⎢
⎤
⎦⎥ = α0 | 0〉 +α1 |1〉
|ϕ〉 = σ1 |ψ 〉 = 0 1
1 0⎡⎣⎢
⎤⎦⎥α0
α1
⎡
⎣⎢
⎤
⎦⎥ =
α1
α0
⎡
⎣⎢
⎤
⎦⎥ = α1 | 0〉 +α0 |1〉
|ϕ〉 = σ 2 |ψ 〉 = 0 −i
i 0⎡⎣⎢
⎤⎦⎥α0
α1
⎡
⎣⎢
⎤
⎦⎥ = i
−α1
α0
⎡
⎣⎢
⎤
⎦⎥ = −iα1 | 0〉 + iα0 |1〉
|ϕ〉 = σ 3 |ψ 〉 = 1 0
0 −1⎡⎣⎢
⎤⎦⎥α0
α1
⎡
⎣⎢
⎤
⎦⎥ =
α0
−α1
⎡
⎣⎢
⎤
⎦⎥ = α0 | 0〉 −α1 |1〉
12
Hadamard Operator• This Operator is Commonly used to
Maximize Superposition of a Qubit in a Basis State
• Example: H =
12
1 11 −1⎡⎣⎢
⎤⎦⎥
|ψ 〉 = α0 | 0〉 +α1 |1〉
H |ψ 〉 =
12
1 11 −1⎡⎣⎢
⎤⎦⎥α0
α1
⎡
⎣⎢
⎤
⎦⎥ =
α0
2(| 0〉+ |1〉) +
α1
2(| 0〉− |1〉)
Hadamard Operator• This Operator is Commonly used to
Maximize Superposition of a Qubit in a Basis State
• Example: H =
12
1 11 −1⎡⎣⎢
⎤⎦⎥
|ψ 〉 = 0 | 0〉 +1|1〉 =|1〉
H |ψ 〉 =
12
1 11 −1⎡⎣⎢
⎤⎦⎥
01⎡⎣⎢⎤⎦⎥= 1 / 2
−1 / 2
⎡
⎣⎢⎢
⎤
⎦⎥⎥= (1 / 2) | 0〉 − (1 / 2) |1〉
Prob[| 0〉 measured] = (1 / 2)2 = 50%
Prob[|1〉 measured] = (1 / 2)2 = 50%
13
Beam Splitter• 50-50 Beam Splitter Performs a Hadamard
Transform on Particles (location/spatially encoded information)
• Beam Splitters have been Constructed for Quantum Particles other than Photons
|ϕ〉 = H |ψ 〉 =
12
α0 +α1
α0 −α1
⎡
⎣⎢
⎤
⎦⎥
|ψ 〉 = α0 | 0〉 +α1 |1〉 | 0〉
|1〉
H
Single Qubit Pauli Operators
Pauli-X
Pauli-Y
Pauli-Z (π/2 gate)
0 11 0
⎡
⎣⎢
⎤
⎦⎥
0 −ii 0
⎡
⎣⎢
⎤
⎦⎥
1 00 −1
⎡
⎣⎢
⎤
⎦⎥
X
Y
Z
14
Other Single Qubit Operations
Hadamard
Phase(π/4 gate)
“T” (π/8 gate)
12
1 11 −1
⎡
⎣⎢
⎤
⎦⎥
1 00 i
⎡
⎣⎢
⎤
⎦⎥
1 00 eiπ 4
⎡
⎣⎢⎢
⎤
⎦⎥⎥
H
S
T
Probability Amplitude Single Qubit Rotations
cosθ2
−isinθ2
−isinθ2
cosθ2
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥ R x (θ )
R y (θ )
R z (θ )
cosθ2
−isinθ2
sinθ2
cosθ2
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
e− iθ 2 00 eiθ 2
⎡
⎣⎢⎢
⎤
⎦⎥⎥
15
Single Qubit Operations (Square Root of X)
12
eiπ 4 e− iπ 4
e− iπ 4 eiπ 4
⎡
⎣⎢⎢
⎤
⎦⎥⎥
4 1 (1 )2
ie iπ = + 4 1 (1 )2
ie iπ− = −
12
eiπ 4 e− iπ 4
e− iπ 4 eiπ 4
⎡
⎣⎢⎢
⎤
⎦⎥⎥= 1
21+ i 1− i1− i 1+ i
⎡
⎣⎢
⎤
⎦⎥
12
⎛⎝⎜
⎞⎠⎟
21+ i 1− i1− i 1+ i
⎡
⎣⎢
⎤
⎦⎥
2
= 14
0 44 0
⎡
⎣⎢
⎤
⎦⎥ =
0 11 0
⎡
⎣⎢
⎤
⎦⎥
Transformation Matrix:
From Euler’s Identity:
Two Gates in Series (Square of Matrix):
Single Qubit Operations (Square Root of Pauli
Operators)
• Square Root of X (NOT)
• Square Root of Y:
• Square Root of Z:
V = X = eiπ4
1 −1−1 1
⎡
⎣⎢
⎤
⎦⎥= 1
2e
iπ4 e
− iπ4
e− iπ
4 eiπ4
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥= 1
21+ i 1− i1− i 1+ i
⎡
⎣⎢
⎤
⎦⎥
Y = eiπ4
1 i− i 1
⎡
⎣⎢
⎤
⎦⎥= 1
2e
iπ4 −e
iπ4
eiπ4 e
iπ4
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥= 1
21+ i −1− i1+ i 1+ i
⎡
⎣⎢
⎤
⎦⎥
Z = e
iπ4
0 00 2
⎡
⎣⎢
⎤
⎦⎥= 1 0
0 i⎡
⎣⎢
⎤
⎦⎥
16
Multi Qubit Systems (Circuits)• Multi qubit systems are represented in
terms of a “product quantum state” • Consider a System of Two qubits, the state
of this system is a superposition of:
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
δχβα Amplitude for 00
Amplitude for 01 Amplitude for 10 Amplitude for 11