Criteria for universality of quantum gates

TL;DR

This paper presents a simple, finite-step algorithm to determine whether a set of quantum one-qudit gates is universal and identifies how to augment non-universal sets to achieve universality.

Contribution

It introduces compact criteria and an algorithm for deciding the universality of quantum gate sets and guides how to modify non-universal sets for universality.

Findings

Provides a finite-step decision algorithm for universality.

Characterizes gate sets that can be made universal by adding specific gates.

Offers a practical method for designing universal quantum gate sets.

Abstract

We consider the problem of deciding if a set of quantum one-qudit gates $\mathcal{S}=\{U_1,\ldots,U_n\}$ is universal. We provide the compact form criteria leading to a simple algorithm that allows deciding universality of any given set of gates in a finite number of steps. Moreover, for a non-universal $\mathcal{S}$ our criteria indicate what type of gates can be added to $\mathcal{S}$ to turn it into a universal set.