A Lie-algebraic Criterion for the Universality of Exponentiated Quantum Gates

TL;DR

This paper introduces a Lie algebraic criterion and a polynomial-time algorithm to determine the universality of exponentiated qudit gates, linking quantum control to Lie algebra representation theory.

Contribution

It formulates a new Lie algebraic approach to decide gate universality and proves that two generators suffice for universal control of qudits.

Findings

A polynomial-time algorithm for universality decision based on Lie algebraic criteria.

Nonuniversality is characterized by invariant subspaces and graph connectivity.

Two generators are sufficient for universal control of qudits.

Abstract

We present a criterion that serves as the basis for a polynomial-time algorithm to decide whether a finite set of qudit gates exponentiated by some Hamiltonians is universal. Our approach formulates universality in Lie algebraic terms and applies Borel--de Siebenthal theory with a diagonal generator having incommensurate spectrum. In this framework, nonuniversality is detected by invariant subspaces, equivalently by a graph-connectivity obstruction, while universality is repaired by adding generators that couple disconnected components. We further prove that two generators are sufficient for universal control. Our work reveals a profound link between qudit universality and irreducibility of Lie algebra representations.