Universality of single qudit gates

TL;DR

This paper provides a criterion and an algorithm to determine whether a set of single-qudit quantum gates is universal, based on their adjoint representations and group-theoretic properties, with implications for quantum computing.

Contribution

The authors introduce a simple, finite-step algorithm to decide universality of single-qudit gates using Lie group and algebra techniques, and establish a general classification theorem.

Findings

A necessary condition for universality involves commutation properties of adjoint representations.

An additional distance criterion ensures universality when met.

A practical algorithm for universality decision is proposed.

Abstract

We consider the problem of deciding if a set of quantum one-qudit gates $\mathcal{S}=\{g_1,\ldots,g_n\}\subset G$ is universal, i.e if the closure $\overline{<\mathcal{S}>}$ is equal to $G$, where $G$ is either the special unitary or the special orthogonal group. To every gate $g$ in $\mathcal{S}$ we asign its image under the adjoint representation $\mathrm{Ad}_g$, where $\mathrm{Ad}:G\rightarrow SO(\mathfrak{g})$ and $\mathfrak{g}$ is the Lie algebra of $G$. The necessary condition for the universality of $\mathcal{S}$ is that the only matrices that commute with all $\mathrm{Ad}_{g_i}$'s are proportional to the identity. If in addition there is an element in $<\mathcal{S}>$ whose Hilbert-Schmidt distance from the centre of $G$ belongs to $]0,\frac{1}{\sqrt{2}}]$, then $\mathcal{S}$ is universal. Using these we provide a simple algorithm that allows deciding the universality of any set of $d$-dimensional gates in a finite number of steps and formulate the general classification theorem.