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Optimal SynthesisOptimal Synthesisof Linear Reversible Circuitsof Linear Reversible Circuits
Ketan Patel, Igor Markov, and John Hayes
University of MichiganElectrical Engineering & Computer Science
DARPADARPA
OutlineOutline
• Motivation
• Background
• Lower Bound
• Synthesis Algorithm
• Application: [Quantum] Stabilizer Circuits
• Future work
MotivationMotivation
Reversible Computation
• “Energy-free” computation
• Reversible applications: cryptography, DSP, etc.
Quantum Computation
• Application to important quantum circuits: stabilizer circuits
CC--NOT GateNOT Gate
First input (control) passes through unchanged
Second input (target) inverted if control=1
C-NOT is linear (under bitwise XOR operation ⊕)
i.e.
C-NOT GateInput Output
00 0001 0110 1111 10
nxxfxfxxf }1,0{, xallfor )()()( 212121 ∈⊕=⊕
CC--NOT CircuitsNOT Circuits
Compute linear transformation over {0,1}n
Mapping of determine full mapping
Example:
C-NOT Circuit
x1x2x3x4
y1y2y3y4
0000|0000| →
1000|,0100|,0010|,0001|
)0100(|)1000(|)1100(| fff ⊕=
Matrix RepresentationMatrix Representation
Can use matrix representation
Example:
C-NOT action corresponds to multiplication by elementary matrix
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡•⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡→
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⇔
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
→→→
110
011
101110100
011| 101110100
101|100|110|010|100|001|
C-NOT Gate
⇔⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
110010001
column (control)
row (target)
Matrix Representation (cont.)Matrix Representation (cont.)
Concatenation of C-NOT’s corresponds to matrix multiplication
Example:
Matrix row reduction ⇒ decomposition into elementary matrices
(recall Gaussian elimination)
G1 G2 G3434214342143421
123 G
100110001
G
101010001
G
100010101
101110100
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡•
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡•
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⇒
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡•
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡•
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
100110001
101010001
100010101
C-NOT Synthesis binary matrix row reduction# of gates = # of row operations
Example:
Gaussian Elimination Requires O(n2) row ops (gates)
CC--NOT SynthesisNOT Synthesis
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⇒
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⇒
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⇒
→→→
100010001
100110001
101110001
233113
3 2 1 3 3 1
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
101110100
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
101110100
⇒
Number of n-wire linear reversible transformations
If all transformations require ≤ d gates, then the number of d-gate circuits must be no smaller
⇒ c n2/log n ≤ d for some c>0Conclusion: need at least c n2/log n gates
Lower BoundLower Bound
)22(1
0∏ −−
=
n
i
in
dn
i
in nn )1)1(()22(1
0+−≤∏ −
−
=
OurOur Synthesis Algorithm (intuition)Synthesis Algorithm (intuition)
Assume n is large
• Use ≤ n row ops to eliminate duplicate sub-rows
• Leave no more than 3 non-zero sub-rows relatively few operations necessary to clear
Reduces number of ops by factor ~2⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
10001101
1001101101
L
L
L
L
MMM
L
L
L
L
L
n
• Group columns into sections of size ≤ αlog n with α <1• Eliminate duplicate sub-rows in each column section
# row ops = O(n) • # sections = O(n2/log n)• Place 1s on the diagonal
# row ops = O(n)• Eliminate (relatively few) remaining off-diagonal 1s
# row ops = O(2 α log n) • # sections = O(nα) • n/(α log n) =O(n1+α/log n)
Total # of row operations: O(n2/log n)
Synthesis AlgorithmSynthesis Algorithm
Execution TimeExecution Time
Execution time dominated by row ops⇒ O(n3/log n)
Execution time for Gaussian Elimination⇒ O(n3)
...but how soon do the asymptotics kick in?
Empirical ResultsEmpirical Results
Gaussian Elimin.Algorithm 1
wires
circ
uit s
ize
• Important class of quantum circuits
• Composed of Hadamard, Phase, and C-NOT gates
• Can use blocks of H, P & C gates to synthesize (Aaronson & Gottesman)
Circuit size dominated by size of C-blocks
Application: Stabilizer CircuitsApplication: Stabilizer Circuits
H P
H P P P
H P P P
C P H P C P
ReRelatedlated TopicsTopics
• Optimal column partitioning
• Synthesis of general reversible circuits
– T-C-N-T decomposition
• Applications to synthesis of quantum circuits
– E.g., via the Gottesman-Knill theorem
• Reducing circuit depth rather than size
Thank you
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