Learning quantum symmetries
Abstract
Quantum algorithms are powerful tools for finding symmetries of classical objects, most famously through Shor's algorithm and the Hidden Subgroup Problem (HSP). In this work, we study quantum algorithms for learning symmetries of \emph{quantum} objects. Existing work in this area centres on the recently introduced State Hidden Subgroup Problem (StateHSP), a quantum generalisation of HSP in which the task is to learn the symmetry subgroup of a quantum state. We develop efficient quantum algorithms for non-abelian StateHSP when the hidden subgroup is normal and the ambient group belongs to a broad class of non-abelian groups, extending the previous general theory beyond the abelian setting. StateHSP learns \emph{Bose} symmetries, under which a state must be invariant exactly under the action of the symmetry group. In quantum mechanics, however, physically equivalent pure states are defined only up to global phase. Motivated by this, we introduce a natural notion of \emph{Anyonic} state symmetry learning, based on invariance up to global phase. We give an efficient quantum algorithm by reducing the problem to StateHSP, where the reduction rests on a new correspondence between linearisations of projective representations and linear error-correcting codes. As an application, we obtain an improved algorithm for learning stabiliser groups of mixed qudit states of arbitrary local dimension. Finally, we introduce symmetry learning problems for other quantum objects, including unitaries, Hamiltonians, and finite collections of states, and give efficient algorithms for them by reduction to state symmetry learning. Together, these results broaden the scope of state symmetry learning as a common algorithmic primitive for learning quantum symmetries.
Related
- Bootstrapping symmetries in quantum many-body systems from the cross spectral form factor
- Phases of flux vacua and their quantum symmetries
- Large Deviation Functions of Open Quantum Systems with Strong Symmetries
- PhD thesis: Modes, States, and Symmetries in quantum Optics for quantum Information and Metrology
- Monopoles, Scattering and Generalized Symmetries on a Quantum Computer
BibTeX
@misc{holt2026learning,
title = {{Learning quantum symmetries}},
author = {Isaac Holt and Sathyawageeswar Subramanian},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38085},
eprint = {2609.38085},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38085},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Learning quantum symmetries
AU - Isaac Holt
AU - Sathyawageeswar Subramanian
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.38085
UR - https://arxiv.org/abs/2609.38085
AB - Quantum algorithms are powerful tools for finding symmetries of classical objects, most famously through Shor's algorithm and the Hidden Subgroup Problem (HSP). In this work, we study quantum algorithms for learning symmetries of \emph{quantum} objects. Existing work in this area centres on the recently introduced State Hidden Subgroup Problem (StateHSP), a quantum generalisation of HSP in which the task is to learn the symmetry subgroup of a quantum state. We develop efficient quantum algorithms for non-abelian StateHSP when the hidden subgroup is normal and the ambient group belongs to a broad class of non-abelian groups, extending the previous general theory beyond the abelian setting. StateHSP learns \emph{Bose} symmetries, under which a state must be invariant exactly under the action of the symmetry group. In quantum mechanics, however, physically equivalent pure states are defined only up to global phase. Motivated by this, we introduce a natural notion of \emph{Anyonic} state symmetry learning, based on invariance up to global phase. We give an efficient quantum algorithm by reducing the problem to StateHSP, where the reduction rests on a new correspondence between linearisations of projective representations and linear error-correcting codes. As an application, we obtain an improved algorithm for learning stabiliser groups of mixed qudit states of arbitrary local dimension. Finally, we introduce symmetry learning problems for other quantum objects, including unitaries, Hamiltonians, and finite collections of states, and give efficient algorithms for them by reduction to state symmetry learning. Together, these results broaden the scope of state symmetry learning as a common algorithmic primitive for learning quantum symmetries.
ER -