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arXiv · 2604.25971

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

Yinuo Xue·Qian Chen·Jing-Song Huang
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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.

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Yinuo Xue, Qian Chen, Jing-Song Huang. 2026-04-28. A Lie-algebraic Criterion for the Universality of Exponentiated Quantum Gates. https://arxiv.org/abs/2604.25971

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