A note on "eight-vertex" universal quantum gates

Abstract

Many well-known and well-studied four by four universal quantum logic gates in the literature are of a specific form, the so called eightvertex form (1.1) [4,5], or similar to it.We present a formalism for universal quantum logic gates of such a form. First, we provide explicit formulas in terms of matrix entries, which are the necessary and sufficient conditions for such a matrix to be a solution to the Yang-Baxter equation (2.2). Then, combining this with the conditions needed for being unitary (2.5) and being entangling (2.6), we give a full description of entangling unitary solutions to the Yang-Baxter equation (hence, universal quantum logic gates) of such a specific form. We investigate in detail all the possible cases where some of the eight main entries might or might not be zero.