Obstructions to universality in globally controlled qubit graphs

TL;DR

This paper disproves a conjecture about the conditions for universality in globally controlled qubit graphs, revealing hidden symmetries beyond graph automorphisms that affect quantum control.

Contribution

It provides explicit counterexamples and demonstrates that graph automorphisms do not fully determine the universality of control in quantum systems.

Findings

Counterexamples with trivial automorphism groups are not universal.

Hidden symmetries beyond automorphisms influence control universality.

Breaking graph symmetries does not always ensure universality.

Abstract

Global control offers a promising route to scalable quantum computing. A recent conjecture by Hu et al. (arXiv:2508.19075) proposes that any connected qubit graph equipped with global Ising-type interactions and tunable global transverse fields achieves universality if and only if an additional control field breaks every non-trivial automorphism of the underlying graph. We disprove this conjecture by exhibiting explicit seven- and nine-qubit counterexamples: connected graphs with trivial automorphism group for which the generated Lie algebra is nonetheless not universal. Our analysis reveals that graph automorphisms capture only part of the Hamiltonian symmetry structure: there exist hidden symmetries beyond the automorphism group of the graph. Additionally, in the case of non-trivial automorphism group, we find control terms which break the graph symmetries but are still not universal. These findings sharpen the characterization of universality for globally controlled quantum systems.