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Recognizing Small-Circuit Structure in Two-Qubit Operators and Timing Hamiltonians to Compute Controlled-Not Gates

Vivek V. Shende·Stephen S. Bullock·Igor L. Markov
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Abstract

This work proposes numerical tests which determine whether a two-qubit operator has an atypically simple quantum circuit. Specifically, we describe formulae, written in terms of matrix coefficients, characterizing operators implementable with exactly zero, one, or two controlled-not (CNOT) gates and all other gates being one-qubit. We give an algorithm for synthesizing two-qubit circuits with optimal number of CNOT gates, and illustrate it on operators appearing in quantum algorithms by Deutsch-Josza, Shor and Grover. In another application, our explicit numerical tests allow timing a given Hamiltonian to compute a CNOT modulo one-qubit gates, when this is possible.

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Vivek V. Shende, Stephen S. Bullock, Igor L. Markov. 2003-12-31. Recognizing Small-Circuit Structure in Two-Qubit Operators and Timing Hamiltonians to Compute Controlled-Not Gates. https://doi.org/10.1103/physreva.70.012310

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