Efficient decomposition of single-qubit gates intoVbasis circuits

Abstract

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 $\le 12{\log }_5(2/\epsilon )$. 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. DOI: http://dx.doi.org/10.1103/PhysRevA.88.012313 ©2013 American Physical Society