This work proposes numerical tests which determine whether a two-qubitoperator has an atypically simple quantum circuit. Specifically, we describeformulae, written in terms of matrix coefficients, characterizing operatorsimplementable with exactly zero, one, or two controlled-not (CNOT) gates andall other gates being one-qubit. We give an algorithm for synthesizingtwo-qubit circuits with optimal number of CNOT gates, and illustrate it onoperators appearing in quantum algorithms by Deutsch-Josza, Shor and Grover. Inanother application, our explicit numerical tests allow timing a givenHamiltonian to compute a CNOT modulo one-qubit gates, when this is possible.
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