Symmetry-initialized quantum Gibbs sampling: a non-Abelian asymmetry cascade

2026-07-27

AI Summary

This work addresses the mixing bottleneck faced by quantum Gibbs samplers under weak symmetry breaking by proposing an acceleration framework grounded in group representation theory. By constructing an asymmetry cascade mechanism within non-Abelian groups, the study reveals that matching only the first moment is insufficient to eliminate slow-mode overlap; instead, a projection operation based on group-averaged asymmetry is required. The approach integrates representation theory, group-invariant input construction, subgroup lattice indexing, and the Davies master equation model. When applied to an SU(2) Davies sampler, it achieves a relaxation rate that scales linearly with the symmetry-breaking parameter, substantially accelerating mixing. The method also rigorously characterizes the range of asymmetric target states for which effective acceleration is guaranteed.

Abstract

For a quantum Gibbs sampler whose mixing bottleneck is a weakly broken symmetry, I show that the correct initialization is determined by representation theory. I first prove a general speedup-versus-prefactor dichotomy: by exactly eliminating slow-mode overlap, I convert a nominal prefactor reduction into a fundamental, asymptotic acceleration of the system's mixing time. I then show that when the bottleneck is a weakly broken symmetry, the otherwise exponentially expensive bottleneck eigenvector is the symmetry charge.For a non-Abelian group $G$, I prove that the correct initialization target is the group-averaging asymmetry. Projecting this asymmetry out requires a $G$-invariant input. Partial, subgroup-invariant inputs produce a cascade of mixing-time speedups indexed by the subgroup lattice. Conversely, matching first moments alone is provably insufficient.The asymmetry is a directly measurable initialization target. I verify these results analytically and numerically in an $SU(2)$ Davies sampler where total-spin multiplets constitute the slow modes. The predicted speedup cascade emerges with relaxation rates scaling linearly with the symmetry-breaking parameter. Finally, the model maps the boundaries of the asymmetry target, illustrating where the separate matching of conserved logical data becomes necessary.

Problem

• quantum Gibbs sampling
• weakly broken symmetry
• mixing time
• non-Abelian group
• symmetry charge

Innovation

• quantum Gibbs sampling
• symmetry breaking
• non-Abelian group
• mixing time acceleration
• asymmetry cascade