Efficient Decomposition of Single-Qubit Gates into V Basis Circuits

We develop efficient algorithms for compiling single-qubit unitary gates into circuits over the universal \(V\) basis. The \(V\) basis is an alternative universal basis to the more commonly studied basis consisting of Hadamard and \(\pi /8\) gates. We propose two classical algorithms for quantum circuit compilation: the first algorithm has expected polynomial time [in precision \(log(1/\epsilon )\)] and produces an \(\epsilon\) approximation to a single-qubit unitary with a circuit depth \(\leq 12 {log}_5(2/\epsilon )\). The second algorithm performs optimized direct search and yields circuits a factor of 3 to 4 times shorter than our first algorithm, but requires time exponential in \(log(1/\epsilon )\); however, we show that in practice the runtime is reasonable for an important range of target precisions. Decomposing into the \(V\) basis may offer advantages when considering the fault-tolerant implementation of quantum circuits.