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

For a quantum Gibbs sampler whose mixing bottleneck is a weakly broken symme- try, I show that the right initialization is determined by representation theory. I first prove a speedup-versus-prefactor dichotomy: nulling the slow mode’s overlap exactly turns an approximate prefactor saving into a genuine mixing-time speedup. I then show that when the bottleneck is a weakly broken symmetry the otherwise exponentially ex- pensive bottleneck eigenvector is the symmetry charge, known a priori at no cost. For a non-Abelian group G the correct initialization target is the group-twirl asymmetry; nulling it requires a G-invariant input, and partial, subgroup-invariant inputs produce a cascade of mixing-time speedups indexed by the subgroup lattice—so that matching first moments alone is provably insufficient. The asymmetry is a directly measurable initialization target. I close with a fully verified SU(2) Davies sampler in which the slow modes are the total-spin multipoles, the predicted cascade appears with rates scaling linearly in the breaking, and the limits of the asymmetry target (the separate matching of conserved logical data) are exhibited explicitly.