controlled v gates*mitch/class/8381/week12/quanlog2v2.pdf · 2019-04-01 · 2 fredkin gate •...
TRANSCRIPT
1
V
Controlled-V
1 0 0 00 1 0 0
0 012+
12
i 12−
12
i
0 012−
12
i 12+
12
i
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥
1 0 0 00 1 0 0
0 012−
12
i 12+
12
i
0 012+
12
i 12−
12
i
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥
V+
Controlled-V+
Controlled V Gates*
Quantum Circuits• Cascades of Quantum Gates • Each”Quantum Wire” Represents a Qubit • Quantum Wires Represent the Transform of Qubits
in Time or Space • Quantum Gates Represent the Evolution or
Transformation of a Qubit • Qubits have been Implemented as:
– photons with polarization indicating state – electrons with spin state – trapped ions with energy level states – NMR (nuclear magnetic resonance) pulses – Superconducting phenomena (josephson junctions) – others
2
Fredkin Gate• 3-Input/Output Gate • Classical Version is Logically Reversible and Physically
Irreversible Defined by Following Truth Table • Quantum Version is Fully Reversible
a b c a’ b’ c’0 0 0 0 0 00 0 1 0 0 10 1 0 0 1 00 1 1 0 1 11 0 0 1 0 01 0 1 1 1 01 1 0 1 0 11 1 1 1 1 1
Fredkin (controlled-swap) Gatex1
x2
x1
y1 = x1 x2 + x1 x3
x3 y2 = x1 x2 + x1 x3
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 0 1 00 0 0 0 0 1 0 00 0 0 0 0 0 0 1
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
Classical Fredkin Gate
|ψ a 〉
|ψ b 〉
|ψ c 〉
|ψ 'a 〉
|ψ 'b 〉
|ψ 'c 〉
3
Quantum Fredkin Gate• Quantum Fredkin Gate is 3-Qubit Quantum
System • 8-Dimensional Complex Vector Space with Basis:
• Tensor Product of Three Single-Qubit State Vectors:
| 000〉,| 001〉,| 010〉,| 011〉,|100〉,|101〉,|110〉,|111〉
ab⎡⎣⎢
⎤⎦⎥⊗ c
d⎡⎣⎢
⎤⎦⎥⊗ e
f⎡⎣⎢
⎤⎦⎥=
acadbcbd
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
⊗ ef
⎡⎣⎢
⎤⎦⎥=
aceacfadeadfbcebcfbdebdf
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
Quantum Fredkin Gate• Basis Vectors in 2-D Hilbert Space: • 8-D Hilbert Space Basis:
| 0〉 = 10⎡⎣⎢⎤⎦⎥
|1〉 = 01⎡⎣⎢⎤⎦⎥
| 000〉 =
10000000
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
| 001〉 =
01000000
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
| 010〉 =
00100000
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
| 011〉 =
00010000
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
|100〉 =
00001000
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
|101〉 =
00000100
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
|110〉 =
00000010
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
|111〉 =
00000001
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
4
8-D Hilbert Space Basis• Note that Basis Vectors are Column(Row) Vectors
of the Identity Transform:
• Can Construct Matrix Representation of Linear Operator as:
I8 =
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
GFredkin =| 000〉〈000 | + | 001〉〈001| + | 010〉〈010 | + | 011〉〈011| + |100〉〈100 | + |101〉〈110 | + |110〉〈101| + |111〉〈111|
Fredkin Gate Transfer Matrix• Computing Expression:
• Yields the Fredkin Transfer Matrix:
GFredkin =
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 0 1 00 0 0 0 0 1 0 00 0 0 0 0 0 0 1
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
GFredkin =| 000〉〈000 | + | 001〉〈001| + | 010〉〈010 | + | 011〉〈011| + |100〉〈100 | + |101〉〈110 | + |110〉〈101| + |111〉〈111|
Note: Book (p. 154) uses Different Qubit for Control so Matrix Differs
5
Toffoli Gatex1
x2
y
x1
x2
x1x2⊕ y
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 0 10 0 0 0 0 0 1 0
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
Classical Toffoli Gate
|ψ 〉
|ϕ〉
|ζ 〉
|ψ '〉
|ϕ '〉
|ζ '〉
Toffoli Gate Transfer Matrix• Computing Expression:
• Yields the Toffoli Transfer Matrix:
GToffoli =
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 0 10 0 0 0 0 0 1 0
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
GToffoli =| 000〉〈000 | + | 001〉〈001| + | 010〉〈010 | + | 011〉〈011| + |100〉〈100 | + |101〉〈110 | + |110〉〈111| + |111〉〈110 |
6
Generalized Toffoli Gate
|x1〉
|x2〉
|y〉
|x’1〉
|x’2〉
|y’〉
|x3〉 |x’3〉
3 or More Control Qubits for a NOT Operation
Quantum Computers• Quantum Particles Used to Store Information • Can Simulate Complex Physical Systems
with n-qubit QC • What is a QC?
– According to Steane: “the quantum computer is first and foremost a machine, which is a theoretical construct, like a thought experiment, whose purpose is to allow quantum information processing to be formally defined”
7
Quantum Computers• What is a Quantum Computer?
– According to D. Deutsch: “A quantum computer is a set of n qubits in which the following operations are feasible”
1. Each qubit can be prepared in some known state. 2. Each qubit can be measured in the basis {|0〉, |1〉}. 3. A universal quantum gate (or set of gates) can be
applied at will to any fixed-size subset of qubits. 4. The qubits do not evolve other than via the above
transformations.
Quantum Circuits• Interconnection of Quantum Gates in a
Serial Cascade • Circuits are Acyclic: Feedback from
One Part of Circuit to Another Not Allowed
• There is no Fanout-Due to the No Cloning Theorem
• These Restrictions Ensure Physical Reversibility
8
Half Adder Circuit
• Produces Sum and Carryout When Input Qubits are in Basis States
| a〉
| b〉
| 0〉
| Garbage〉
| Sum〉
| CarryOut〉
Ancilla Qubit
Full Adder Example• From Benchmark Page of Dmitri Maslov
– http://www.cs.uvic.ca/~dmaslov/FULL ADDER CIRCUIT
ancilla bit
garbage bits
9
Qubit Swapping Circuit• Swaps a and b
| a〉
| b〉
| b〉
| a〉
| b ' = a⊕ b〉
| a '〉 =| a〉 | a ''〉 =| a '⊕ b '〉 =| a⊕ (a⊕ b)〉
| b '' = a⊕ b〉
| a '''〉 =| b〉
| b ''' = a ''⊕ b ''〉
hwb5 Circuit• From Benchmark Page of Dmitri Maslov
– http://www.cs.uvic.ca/~dmaslov/hidden weighted bit function (5 bits)
How big is the transformation matrix?
10
Decoder Circuit• From Benchmark Page of Dmitri Maslov
– http://www.cs.uvic.ca/~dmaslov/2 to 4 decoder with enable
Cascade of 3 Fredkin Gates What is the overall transformation matrix?
Example Circuit|x1〉
|x2〉
|y〉
|x1〉
| x1 ⊕ x2〉
| x1 x2 ⊕ y〉
U1 =
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 0 10 0 0 0 0 0 1 0
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
U2 =
1 0 0 00 1 0 00 0 0 10 0 1 0
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
⊗1 00 1⎡
⎣⎢
⎤
⎦⎥
11
Kronecker Product and Properties 11 12 1
21 22 2
1 2
n
n
m m mn
a B a B a Ba B a B a B
A B
a B a B a B
⎡ ⎤⎢ ⎥⎢ ⎥⊗ =⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
!!
" " # "!
(αA)⊗ B = α( A⊗ B)A⊗ (αB) = α( A⊗ B)( A+ B)⊗C = A⊗C + B⊗CA⊗ (B + C) = A⊗ B + A⊗CA⊗ (B⊗C) = ( A⊗ B)⊗C( A⊗ B)t = At ⊗ Bt
( A⊗ B)−1 = A−1 ⊗ B−1
( A⊗ B)(C ⊗ D) = AC ⊗ BD
Mixed Product Rule: Matrices Must be of Appropriate Dimension
Also Known as: 1) Tensor Matrix Product 2) Direct Matrix Product
Example Circuit
U1 =
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 0 10 0 0 0 0 0 1 0
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
U2 ==
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 1 0 0 00 0 0 0 0 1 0 0
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
|x1〉
|x2〉
|y〉
|x1〉
| x1 ⊕ x2〉
| x1 x2 ⊕ y〉
12
Example Circuit
U =U2U1 =
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 1 0 0 00 0 0 0 0 1 0 0
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 0 10 0 0 0 0 0 1 0
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
Reverse Order
|x1〉
|x2〉
|y〉
|x1〉
| x1 ⊕ x2〉
| x1 x2 ⊕ y〉
Example Circuit
U =U2U1 =
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 1 0 0 00 0 0 0 0 1 0 0
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 0 10 0 0 0 0 0 1 0
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
U =U2U1 =
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 0 0 10 0 0 0 0 0 1 00 0 0 0 1 0 0 00 0 0 0 0 1 0 0
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
13
Example Circuit|x1〉
|x2〉
|x3〉
|x1〉
|x2〉
| x1⊕ x3〉
This is tricky because middle qubit is “skipped” Can Permute network, compute matrix,
then un-permute matrix
Example Circuit|x1〉
|x2〉
|x3〉
|x1〉
|x2〉
| x1⊕ x3〉
U132 =
1 0 0 00 1 0 00 0 0 10 0 1 0
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
⊗1 00 1⎡
⎣⎢
⎤
⎦⎥
|x2〉 | x1x3〉
U132 =
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 1 0 0 00 0 0 0 0 1 0 0
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
14
“Un-Permute” Matrix
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 1 0 0 00 0 0 0 0 1 0 0
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
right-side�|x1x3x2>
left-side �|x1x3x2>
|000>|001>|010>|011>|100>|101>|110> |111>
|000>
|001>
|010>
|011>
|100>
|101>
|110>
|111>
“Un-Permute” Matrix
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 1 0 0 00 0 0 0 0 1 0 0
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
right-side�|x1x2x3>
left-side �|x1x2x3>
|000>|010>|001>|011>|100>|110>|101>|111>
|000>
|010>
|001>
|011>
|100>
|110>
|101>
|111>
15
“Un-Permute” Matrix
1 0 0 0 0 0 0 00 0 1 0 0 0 0 00 1 0 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 0 10 0 0 0 1 0 0 00 0 0 0 0 0 1 0
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
right-side�|x1x2x3>
left-side �|x1x2x3>
|000>|001>|010>|011>|100>|101>|110> |111>
|000>
|010>
|001>
|011>
|100>
|110>
|101>
|111>
“Un-Permute” Matrix
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 1 0 00 0 0 0 1 0 0 00 0 0 0 0 0 0 10 0 0 0 0 0 1 0
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
right-side�|x1x2x3>
left-side �|x1x2x3>
|000>|001>|010>|011>|100>|101>|110> |111>
|000>
|001>
|010>
|011>
|100>
|101>
|110>
|111>
16
• General controlled gates that control some 1-qubit unitary operation U are useful
Quantum Gates*
U
u00 u01
u10 u11
⎛
⎝⎜
⎞
⎠⎟
C(U)
U
C2(U)
U
U
etc.
*from D.M. Miller
Discrete Universal Gate Set • Example 1: Four-member “standard” gate set
Quantum Gates*
1 0 0 00 1 0 00 0 0 10 0 1 0
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ 1
21 11 −1⎛ ⎝ ⎜ ⎞
⎠
H
1 00 i
⎛ ⎝ ⎜ ⎞
⎠
S �/8
1 00 eiπ /4⎛ ⎝ ⎜ ⎞
⎠
CNOT Hadamard Phase �/8 (T) gate
• Example 2: {CNOT, Hadamard, Phase, Toffoli}
*from D.M. Miller
17
Quantum Circuits*
Example: Algebraic analysis
U4U2 U3U1 U5U0
V V† V
=
U
?x1
x2
x3
• Is U0 (x1, x2, x3) T= U5U4U3U2U1(x1, x2, x3) T?
*from D.M. Miller
• Ad hoc designs known for many specific functions and gates.
• Example 1 illustrating a theorem by [Barenco et al. 1995]: Any C2(U) gate can be built from CNOTs, C(V), and C(V†) gates, where V2 = U
Barenco’s Theorem
V V† V
=
U
18
Quantum Circuits*Example (contd)U1 = I1⊗C(V)
=1 00 1
⎛ ⎝ ⎜ ⎞
⎠ ⊗
1 0 0 00 1 0 00 0 v00 v010 0 v10 v11
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
=
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 v00 v01 0 0 0 00 0 v10 v11 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 v00 v010 0 0 0 0 0 v10 v11
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
*from D.M. Miller
Quantum Circuits*Example (contd)
U2 = U4 = CNOT(x1,x2)⊗ I1
=
1 0 0 00 1 0 00 0 0 10 0 1 0
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⊗
1 00 1⎛ ⎝ ⎜ ⎞
⎠ =
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 1 0 0 00 0 0 0 0 1 0 0
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
*from D.M. Miller
19
Quantum Circuits*
Example (contd) – U5 is the same as U1 but has x1and x2 permuted (tricky!) – It remains to evaluate the product of five 8 x 8 matrices
U5U4U3U2U1 using the fact that VV† = I and VV = U1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 v00 v01 0 00 0 0 0 v10 v11 0 00 0 0 0 0 0 v00 v010 0 0 0 0 0 v10 v11
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 1 0 0 00 0 0 0 0 1 0 0
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 v00 v10 0 0 0 00 0 v01 v11 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 v00 v100 0 0 0 0 0 v01 v11
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 1 0 0 00 0 0 0 0 1 0 0
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 v00 v01 0 0 0 00 0 v10 v11 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 v00 v010 0 0 0 0 0 v10 v11
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
=
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 v00v00+ v10v10 v00v01+ v10v110 0 0 0 0 0 v01̀v00+ v11v10 v01v01+ v11v11
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
= U0
*from D.M. Miller
V V † V† V
Quantum Full Adder*Cin
x
y
0
Sum
Cout
Propagate
x
• Low-cost (7 gate) Full Adder • Due to Synthesis Method by Maslov, Young,
Miller, and Dueck 2004
20
Classical Reversible Logic• Unitary Transformation Matrices are Permutation
Matrices
• Fredkin is Universal Gate
• Generalized Toffoli, Feynman, Toffoli are Universal
• Some Investigation for “traditional circuitry” Occurred (adiabatic logic)
• Quality (Cost) of Circuit – Number of Ancilla and Garbage bits
– Number of Gates
– Number of Equivalent 2-qubit Gates
Summary• Synthesis of Reversible and Quantum Logic Circuits
is Very Immature • Complexity of Classical Logic Functions is
Exponential (2n rows in truth table) - Reversible and Quantum logic is 2n × 2n (transformation matrix)
• Testing, Verification Issues are Unsolved • Simulation of Large Circuits is Still Immature • Some Algorithms have Been Formulated to Solve
Problems that are Intractable on Classical Computers – Schor’s Method for Factoring Large Numbers – Database Search using Grover’s Oracle