Quantum Compiling with Approximation of Multiplexors

A quantum compiling algorithm is an algorithm for decomposing (“compiling”) an arbitrary unitary matrix into a sequence of elementary operations (SEO). Suppose Uin is an NB-bit unstructured unitary matrix (a unitary matrix with no special symmetries) that we wish to compile. For NB > 10, expressing Uin as a SEO requires more than a million CNOTs. This calls for a method for finding a unitary matrix that: (1)approximates Uin well, and (2) is expressible with fewer CNOTs than Uin. The purpose of this paper is to propose one such approximation method. Various quantum compiling algorithms have been proposed in the literature that decompose an arbitrary unitary matrix into a sequence of U(2)-multiplexors, each of which is then decomposed into a SEO. Our strategy for approximating Uin is to approximate these intermediate U(2)-multiplexors. In this paper, we will show how one can approximate a U(2)-multiplexor by another U(2)-multiplexor that is expressible with fewer CNOTs.