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 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 session tomorrow : Cambridge, NIST Los Alamos, and Michigan

Classical Models Quantum Models  0-1 strings E.g., one bit {0,1}  Bool. Functions Gates & circuits  Primary outputs  Lin. combinations of 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/computation can 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>)/

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 CNOT gate x y x yxyx

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 exist iff the gate library is universal

Phase can be ignored Gate 1 Gate 2Gate 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 23

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” qubits but 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=AB, 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  K 1, K 2 and  such that U=K 1  K 2  E  E * is diagonal (5 gates)  K 1, K 2 have only real entries  The terms K 1, K 2 and  can be found explicitly  Numerical analysis: polar and spectral decompositions  Reduce K 1 and K 2 to tensor products using entanglers  EUE*=E(AB)E*E  E*E ( CD)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 RzRz RzRz RzRz RzRz

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(AB)E* and E(CD)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

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/ quant-ph/ Optimal results for any AB 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., 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

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