an arbitrary two-qubit computation in 23 elementary gates or less stephen s. bullock and igor l....
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An Arbitrary Two-qubit Computationin 23 Elementary Gates or Less
Stephen S. Bullockand Igor L. Markov
University of Michigan
Departments of Mathematics and EECS
Outline
Introduction A crash-course in quantum circuits
Prior work Synthesis by matrix factorizations
Our contribution Entanglers and disentanglers Recognizing tensor products Circuit decompositions
Conclusions and on-going work
Introduction
Abstract synthesis of logic circuits Input: a function that represents a computation Output: a circuit that implements that function Minimize circuit size
Our focus: two-qubit quantum computation Quantum states are complex vectors Computations and gates are 4x4-unitary matrices
Existing commercial applications Secure communication: quantum key distribution
(circuits are small, but gates are very expensive) Future applications: quantum computing
Quantum Information
Most popular carriers … Polarization of an individual photon Spin of an individual nucleus or electron Energy-level of an individual electron
Common features “Basis” states, e.g., and or and Linear combinations are allowed
|0>+|1>; ||2+||2=1; , are complex
Fundamental change from classical info
Special sessiontomorrow : Cambridge, NISTLos Alamos, and Michigan
Classical Models Quantum Models
0-1 strings
E.g., one bit{0,1}
Bool. Functions Gates & circuits
Primary outputs
Lin. combinationsof 0-1 strings E.g., one qubit
|0>+|1> Matrices
Gates & circuits Probabilistic
measurement Mostly ignored here
From Classical to Quantum
All quantum computations M must be unitary M x M-conjugate-transpose = I M-1 (recall “reversible computations”)
A conventional reversible gate/computationcan be extended by linearity E.g., a quantum inverter swaps |0> and |1> Maps the state (|0>+|1>)/2 to itself
Can apply an inverter on one of two qubits E.g., (|00>+i|11>)/2 (|01>+i|10>)/2
0 11 0
Quantum Circuits
Can apply an inverter on one of two qubits E.g., (|00>+i|11>)/2 (|01>+i|10>)/2
How do we describe this computation? Tensor product: IdentityNOT More generally: AB 0 1 0 0
1 0 0 00 0 0 10 0 1 0
0 0 1 00 0 0 11 0 0 00 1 0 0
CNOT gatex
y
xyx
Elementary Gates Q. Computation
Technology-indep. Synthesis
Input: Unitary 4x4-matrix M Generic quantum computation on 2 qubits
Output: circuit in terms of elem. gates that implements M up to a constant
Minimize: circuit cost E.g., gate count or (gate costs)
Solutions existiff the gate library is universal
Phase can be ignored
Gate 1 Gate 2 Gate 3
Previous Work
Proof of universality is constructive [Barenco et. al `95] in Phys. Rev. A Can be interpreted as a synthesis algorithm However, no attempt to minimize #gates
Can be viewed as matrix factorization [Cybenko `01] M=QR with unitary Q & upper-triangular R
(M unitary R diagonal) We count gates, and the answer is 61
Our Work (1)
Two new synthesis procedures (2 qubits) Based on a different matrix factorization
(related to SVD and KAK decompositions) Also reduce generic synthesis to diag synth Small circuits for diagonal computations Smaller overall circuits than QR (Cybenko)
Constructive worst-case bounds Any circuit in gates or less Any circuit with 15 non-constant gates
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Our Work (2)
Lower bounds two-qubit computations (most of them)
that require at least 17 elementary gates At least 15 non-const gates At least 2 CNOTs
Bounds are not constructive and not tight,except for “15 non-const gates”
We never use “temporary storage” qubitsbut this could lead to smaller gate counts
The Entangler and Disentangler
“Computational basis” |00>,|01>,|10> and |11>
The “entangler” computation maps|00> to (|00>+|11>)/2, etc.
The “disentangler”is E-1=E*
Key lemma If U=AB, then EUE* has only real entries An efficient way to recognize tensor products
Circuits For E and E*
A specific circuit for the entangler E
S=diag(1,i) counts as one elementary gate The Hadamard gate H counts as two E* is implemented by reversing the diagram
Change S to S-1=diag(1,-i)
7 elem. gates
The “canonical decomposition” for 2-qubit computations:
U K1,K2 and such that U=K1K2
EE* is diagonal (5 gates) K1,K2 have only real entries
The terms K1, K2 and can be found explicitly Numerical analysis: polar and spectral decompositions
Reduce K1 and K2 to tensor products using entanglers EUE*=E(AB)E*EE*E(CD)E* A,B,C and D are one-qubit computations: 3 gates each
Note that E and E* are the same for any input
Our (Key) Synthesis Procedure
Rz RzRz
Rz
Details (1)
After the initial “divide-and-conquer”many gate cancellations can be made
This brings down max #gates to 28 Only 15 of them depend on input,
which matches an a priori lower bound Further reductions from the analysis
of E(AB)E* and E(CD)E* Max #gates reduced to However, 19 gates depend on the input
23
Details (2)
The structure of the 23-gate circuit
For additional details, see Our paper in Physical Review A http://xxx.lanl.gov/abs/quant-ph/0211002
Validation of Our Synthesis Algo
Implementation in C++ Produces 23 gates for randomly-
generated 4x4-unitary matrices Can capture structure:
several examples in quant-ph/0211002 Optimal results for any AB circuit
(QR decomposition typically 61 gates) For 2-qubit Fourier transform:
a circuit with minimal # of CNOT gates
Conclusions and On-going Work
First generic synthesis algorithm to capture circuit structure, e.g., AB
On-going work Lower and upper bounds of gates (almost done) Solved synthesis of n-qubit diagonal computations
(produce circuits within 2x from optimal)
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Thank you!
Questions are welcome
Quantum dots Nuclear Magnetic Resonance
Ion Traps
Thank you!