Optimising the Solovay-Kitaev algorithm

The Solovay-Kitaev algorithm is the standard method used for approximating arbitrary single-qubit gates for fault-tolerant quantum computation. In this paper we introduce a technique called "search space expansion", which modifies the initial stage of the Solovay-Kitaev algorithm, increasing the length of the possible approximating sequences but without requiring an exhaustive search over all possible sequences. We show that our technique, combined with a GNAT geometric tree search outputs gate sequences that are almost an order of magnitude smaller for the same level of accuracy. This therefore significantly reduces the error correction requirements for quantum algorithms on encoded fault-tolerant hardware.