From Heat to Homology: Spectral Gap Transfer for Exact Quantum Gibbs Sampling at All Temperatures
Abstract
Preparing Gibbs states through dissipative dynamics requires controlling convergence for the chosen Hamiltonian H and coupling operators. However, efficient implementation requires convergence guarantees from the spectral gap of the generator of the dynamics, which remains a crucial challenge. Recent results address the challenge by restricting the dynamics with structural assumptions about the Hamiltonian and its interactions. For the Chen-Kastoryano-Gilyen (CKG) construction and arbitrary finite-dimensional Hamiltonians, we prove an explicit comparison that converts a spectral gap estimate for the infinite-temperature generator into a lower bound on the spectral gap of the CKG quantum Gibbs sampler at every finite positive inverse temperature β. It allows a single spectral gap estimate at infinite temperature to certify convergence across Hamiltonians and temperatures. Such an estimate is often known from a classical spectral gap estimate, such as the classical random walk spectral gap. For Hodge Laplacians of simplicial complexes as Hamiltonians, we construct coupling operators whose generator at infinite temperature has the same spectral gap as a classical simplicial down-up walk. A positive spectral gap of the walk and bounded spectral width W=λmax(H)−λmin(H) give a generator spectral gap independent of the number of simplices at every fixed temperature. We then establish conditions under which the sampler approximately prepares zero-energy (harmonic) states in polynomial evolution time. This provides a conditional convergence guarantee for the state preparation in the thermal approach to quantum topological data analysis.
Related
- Fast mixing of all-to-all quantum systems at high temperatures
- Revealing Many-Body Phases at finite temperatures through Quantum Coherence Distribution
- Nonlinear spectral back transfer limits the temporal coherency of zonal modes
- Improved Prediction of Mössbauer Quadrupole Splitting at High and Low Temperatures with a Quantum Chemical Ensemble Approach
- Holographic shear correlators at low temperatures, and quantum η/s
BibTeX
@misc{leditto2026from,
title = {{From Heat to Homology: Spectral Gap Transfer for Exact Quantum Gibbs Sampling at All Temperatures}},
author = {Caesnan M. G. Leditto and Kuo-Chin Chen and Min-Hsiu Hsieh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02859},
eprint = {2610.02859},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02859},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Heat to Homology: Spectral Gap Transfer for Exact Quantum Gibbs Sampling at All Temperatures
AU - Caesnan M. G. Leditto
AU - Kuo-Chin Chen
AU - Min-Hsiu Hsieh
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02859
UR - https://arxiv.org/abs/2610.02859
AB - Preparing Gibbs states through dissipative dynamics requires controlling convergence for the chosen Hamiltonian H and coupling operators. However, efficient implementation requires convergence guarantees from the spectral gap of the generator of the dynamics, which remains a crucial challenge. Recent results address the challenge by restricting the dynamics with structural assumptions about the Hamiltonian and its interactions. For the Chen-Kastoryano-Gilyen (CKG) construction and arbitrary finite-dimensional Hamiltonians, we prove an explicit comparison that converts a spectral gap estimate for the infinite-temperature generator into a lower bound on the spectral gap of the CKG quantum Gibbs sampler at every finite positive inverse temperature β. It allows a single spectral gap estimate at infinite temperature to certify convergence across Hamiltonians and temperatures. Such an estimate is often known from a classical spectral gap estimate, such as the classical random walk spectral gap. For Hodge Laplacians of simplicial complexes as Hamiltonians, we construct coupling operators whose generator at infinite temperature has the same spectral gap as a classical simplicial down-up walk. A positive spectral gap of the walk and bounded spectral width W=λ_max(H)−λ_min(H) give a generator spectral gap independent of the number of simplices at every fixed temperature. We then establish conditions under which the sampler approximately prepares zero-energy (harmonic) states in polynomial evolution time. This provides a conditional convergence guarantee for the state preparation in the thermal approach to quantum topological data analysis.
ER -