Rapid mixing of Gibbs samplers via quantum Dobrushin--Shlosman conditions
Abstract
Classical Dobrushin--Shlosman theory uses heat-bath block updates to identify sharp temperature thresholds for establishing the rapid mixing of Gibbs samplers. In particular, these mixing guarantees often hold even when stricter single-site conditions fail. We extend this approach to noncommuting quantum lattice systems through a quantum Dobrushin--Shlosman condition for finite-block dynamics, which we call smoothed heat-bath dynamics. We establish this condition in two regimes. For finite-range quantum spin chains at every fixed finite temperature, we prove logarithmic trace-norm mixing in system size, with normalized bounds uniform in on-site fields and single exponential in inverse temperature. We also prove stability under small local perturbations of interacting, possibly noncommuting reference Hamiltonians on graphs of polynomial volume growth. For classical references with bounded local interactions, uniform strong spatial mixing implies our quantum condition. With bounded local terms and fixed physical, geometric and block parameters, the resulting samplers prepare Gibbs states to trace-norm error ε using Npolylog(N/ε) gates and classical operations. Our block updates combine a local Gibbs reset with quantum belief propagation to incorporate interactions across the block boundary, yielding exactly Gibbs-preserving, KMS-reversible channels. Two independently tunable parameters control contraction: an internal evolution time suppresses the influence of sites inside a block, while the block size allows interior contraction to dominate boundary influence. This finite-time relaxation has no counterpart in a classical heat-bath block update.
Related
- Rapid mixing for Gibbs samplers for quantum spin systems with all-to-all interactions
- Spectral Analysis Beyond the Spectral Gap: Rapid Mixing of Quantum Gibbs Samplers in One Dimension
- Hypercontractivity and entropy decay for infinite-dimensional quantum Gibbs samplers
- Dephasing-Enhanced Response and Phase Transitions in Quantum Boltzmann Samplers
- Size-independent Logarithmic Sobolev Inequalities for Quantum Gibbs Samplers
BibTeX
@misc{rouze2026rapid,
title = {{Rapid mixing of Gibbs samplers via quantum Dobrushin--Shlosman conditions}},
author = {Cambyse Rouzé and Daniel Stilck França},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03578},
eprint = {2610.03578},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03578},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rapid mixing of Gibbs samplers via quantum Dobrushin--Shlosman conditions
AU - Cambyse Rouzé
AU - Daniel Stilck França
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03578
UR - https://arxiv.org/abs/2610.03578
AB - Classical Dobrushin--Shlosman theory uses heat-bath block updates to identify sharp temperature thresholds for establishing the rapid mixing of Gibbs samplers. In particular, these mixing guarantees often hold even when stricter single-site conditions fail. We extend this approach to noncommuting quantum lattice systems through a quantum Dobrushin--Shlosman condition for finite-block dynamics, which we call smoothed heat-bath dynamics. We establish this condition in two regimes. For finite-range quantum spin chains at every fixed finite temperature, we prove logarithmic trace-norm mixing in system size, with normalized bounds uniform in on-site fields and single exponential in inverse temperature. We also prove stability under small local perturbations of interacting, possibly noncommuting reference Hamiltonians on graphs of polynomial volume growth. For classical references with bounded local interactions, uniform strong spatial mixing implies our quantum condition. With bounded local terms and fixed physical, geometric and block parameters, the resulting samplers prepare Gibbs states to trace-norm error ε using Npolylog(N/ε) gates and classical operations. Our block updates combine a local Gibbs reset with quantum belief propagation to incorporate interactions across the block boundary, yielding exactly Gibbs-preserving, KMS-reversible channels. Two independently tunable parameters control contraction: an internal evolution time suppresses the influence of sites inside a block, while the block size allows interior contraction to dominate boundary influence. This finite-time relaxation has no counterpart in a classical heat-bath block update.
ER -