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

An Arbitrary Two-qubit Computation in 23 Elementary Gates or Less. Stephen S. Bullock and 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

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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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