A note on "eight-vertex" universal quantum gates

TL;DR

This paper characterizes and classifies eight-vertex form universal quantum gates that are solutions to the Yang-Baxter equation, providing explicit formulas and conditions for unitarity and entanglement.

Contribution

It offers a complete formalism and explicit criteria for eight-vertex form quantum gates to be universal, entangling, and solutions to the Yang-Baxter equation.

Findings

Explicit formulas for eight-vertex gates as solutions to Yang-Baxter

Conditions for unitarity and entanglement of these gates

Classification of cases based on zero or non-zero entries

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 eight-vertex form \eqref{8vertexform} \cite{kaufman etal 05-1,kaufman etal 05-2}, or {\it 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 \eqref{byb}. Then, combining this with the conditions needed for being unitary \eqref{unitarycond} and being entangling \eqref{entanglingcond}, 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.