Quantum circuit synthesis via a random combinatorial search

We use a random search technique to find quantum gate sequences that implement perfect quantum statepreparation or unitary operator synthesis with arbitrary targets. This approach is based on the recent discoverythat there is a large multiplicity of quantum circuits that achieve unit fidelity in performing a given targetoperation, even at the minimum number of single-qubit and two-qubit gates needed to achieve unit fidelity. Weshow that the fraction of perfect-fidelity quantum circuits increases rapidly as soon as the circuit size exceedsthe minimum circuit size required for achieving unit fidelity. This result implies that near-optimal quantumcircuits for a variety of quantum information processing tasks can be identified relatively easily by trying onlya few randomly chosen quantum circuits and optimizing their parameters. In addition to analyzing the casewhere theCNOTgate is the elementary two-qubit gate, we consider the possibility of using alternative two-qubitgates. In particular, we analyze the case where the two-qubit gate is theBgate, which is known to reduce theminimum quantum circuit size for two-qubit operations. We apply the random search method to the problem ofdecomposing the four-qubit Toffoli gate and find a 15-CNOT-gate decomposition