Cyclic groups and quantum logic gates

TL;DR

This paper introduces a new family of universal quantum logic gates derived from cyclic groups, providing both discrete and continuous parameterizations, and explores their relation to the Yang-Baxter equation and Hamiltonian symmetries.

Contribution

It presents a novel formula for an infinite set of universal quantum gates based on cyclic group representations, extending to a continuous family with Yang-Baxterization.

Findings

Derived an infinite family of 4x4 unitary solutions to the Yang-Baxter equation.

Connected discrete cyclic group-based gates to a broader continuous family.

Discussed the Hamiltonian symmetries associated with these quantum gates.

Abstract

We present a formula for an infinite number of universal quantum logic gates, which are $4$ by $4$ unitary solutions to the Yang-Baxter (Y-B) equation. We obtain this family from a certain representation of the cyclic group of order $n$. We then show that this {\it discrete} family, parametrized by integers $n$, is in fact, a small sub-class of a larger {\it continuous} family, parametrized by real numbers $\theta$, of universal quantum gates. We discuss the corresponding Yang-Baxterization and related symmetries in the concomitant Hamiltonian.