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An Arbitrary Two-qubit Computation in 23 Elementary Gates or Less

This paper explores the synthesis of quantum circuits for two-qubit computations using a matrix factorization approach. The authors present two new synthesis procedures and provide constructive worst-case bounds for the number of gates required. They also introduce the concepts of entanglers and disentanglers, which are essential for recognizing tensor products and reducing gate counts. In their synthesis procedure, the authors utilize the canonical decomposition for two-qubit computations and achieve circuits with 23 gates or less.

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An Arbitrary Two-qubit Computation in 23 Elementary Gates or Less

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  1. An Arbitrary Two-qubit Computationin 23 Elementary Gates or Less Stephen S. Bullock and Igor L. Markov University of Michigan Departments of Mathematics and EECS

  2. 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

  3. 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

  4. Quantum Information Special session tomorrow : Cambridge, NIST Los Alamos, and Michigan • 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

  5. 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 • Probabilisticmeasurement • Mostly ignored here

  6. From Classical to Quantum • All quantum computations M must be unitary • MxM-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 1 1 0

  7. 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: IdentityNOT • More generally: AB 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 CNOT gate x x yx y

  8. Elementary Gates Q. Computation

  9. 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

  10. Phase can be ignored Gate 2 Gate 3 Gate 1

  11. 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

  12. 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 23

  13. 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

  14. 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=AB, then EUE*has only real entries • An efficient way to recognize tensor products

  15. 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

  16. Our (Key) Synthesis Procedure • The “canonical decomposition” for 2-qubit computations:  U K1,K2andsuch that U=K1K2 • EE*is diagonal (5 gates) • K1,K2have only real entries • The terms K1, K2andcan be found explicitly • Numerical analysis: polar and spectral decompositions • Reduce K1and K2to tensor products using entanglers • EUE*=E(AB)E*EE*E(CD)E* • A,B,C and Dare one-qubit computations: 3 gates each • Note that E and E* are the same for any input Rz Rz Rz Rz

  17. 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 analysisof E(AB)E* and E(CD)E* • Max #gates reduced to • However, 19 gates depend on the input 23

  18. 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

  19. 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 AB circuit(QR decomposition  typically 61 gates) • For 2-qubit Fourier transform: a circuit with minimal # of CNOT gates

  20. Conclusions and On-going Work • First generic synthesis algorithm to capture circuit structure, e.g., AB • On-going work • Lower and upper bounds of gates (almost done) • Solved synthesis of n-qubit diagonal computations (produce circuits within 2x from optimal) 18

  21. Thank you! Questions are welcome Ion Traps Nuclear Magnetic Resonance Quantum dots

  22. Thank you!

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