Unstructured Unitary Matrices with Fewer CNOTs

A quantum compiler is a software program for decomposing (“compiling”) an arbitrary unitary matrix into a sequence of elementary operations (SEO). The author of this paper is also the author of a quantum compiler called Qubiter. Qubiter uses a matrix decomposition called the Cosine-Sine Decomposition (CSD) that is well known in the field of Computational Linear Algebra. One way of measuring the efficiency of a quantum compiler is to measure the number of CNOTs it uses to express an unstructured unitary matrix (a unitary matrix with no special symmetries). We will henceforth refer to this number as ǫ. In this paper, we show how to improve ǫ for Qubiter so that it matches the current world record for ǫ, which is held by another quantum compiling algorithm based on CSD.