Quantum Gibbs State Preparation via Relative Decoding: From Fast-Mixing Sources to Broader Target Classes
Abstract
Provable guarantees for preparing quantum Gibbs states, such as rapid mixing or gapped coherent preparation paths, are known only for restricted Hamiltonian families and usually must be re-derived for each new family. Building on homomorphic polynomial transduction, a generalization of decoded quantum interferometry (DQI) and Hamiltonian DQI (HDQI), we show how relative decoding transfers such a guarantee from a source Hamiltonian HA to a target HB. Writing each Hamiltonian as a sum of m Pauli terms, the subsets of terms whose product is proportional to the identity, its relations, form a binary linear code, KA for the source and KB for the target, with the Pauli-label matrix as parity-check matrix as in DQI. Starting from the canonical thermofield double (TFD) of HA, a Bell transform, a reversible label map, and a coherent decoder for the quotient code KB/KA prepare an approximate TFD of HB, and hence its Gibbs state. Because this decoder only resolves target relations absent from the source, the reachable inverse temperature is set by the relative distance drel, the fewest terms in any such relation. It can far exceed the ordinary distance dord, the fewest terms in any target relation, which bounds the uniform exact decoding radius of DQI and HDQI. We show that linear drel with efficient decoding at linear radius certifies a constant inverse temperature. As an example, starting from a nearest-neighbor spin chain whose TFD is known to be preparable at every finite temperature, a sparse classical parity-check matrix yields bounded-degree, geometrically nonlocal, noncommuting targets with dord=3 and drel=Θ(m). To our knowledge, this gives a new class of efficiently preparable TFDs.
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BibTeX
@misc{shang2026quantum,
title = {{Quantum Gibbs State Preparation via Relative Decoding: From Fast-Mixing Sources to Broader Target Classes}},
author = {Zhong-Xia Shang and Yufei Wang and Daniel Stilck França},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06667},
eprint = {2610.06667},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06667},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum Gibbs State Preparation via Relative Decoding: From Fast-Mixing Sources to Broader Target Classes
AU - Zhong-Xia Shang
AU - Yufei Wang
AU - Daniel Stilck França
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.06667
UR - https://arxiv.org/abs/2610.06667
AB - Provable guarantees for preparing quantum Gibbs states, such as rapid mixing or gapped coherent preparation paths, are known only for restricted Hamiltonian families and usually must be re-derived for each new family. Building on homomorphic polynomial transduction, a generalization of decoded quantum interferometry (DQI) and Hamiltonian DQI (HDQI), we show how relative decoding transfers such a guarantee from a source Hamiltonian H_A to a target H_B. Writing each Hamiltonian as a sum of m Pauli terms, the subsets of terms whose product is proportional to the identity, its relations, form a binary linear code, K_A for the source and K_B for the target, with the Pauli-label matrix as parity-check matrix as in DQI. Starting from the canonical thermofield double (TFD) of H_A, a Bell transform, a reversible label map, and a coherent decoder for the quotient code K_B/K_A prepare an approximate TFD of H_B, and hence its Gibbs state. Because this decoder only resolves target relations absent from the source, the reachable inverse temperature is set by the relative distance d_rel, the fewest terms in any such relation. It can far exceed the ordinary distance d_ord, the fewest terms in any target relation, which bounds the uniform exact decoding radius of DQI and HDQI. We show that linear d_rel with efficient decoding at linear radius certifies a constant inverse temperature. As an example, starting from a nearest-neighbor spin chain whose TFD is known to be preparable at every finite temperature, a sparse classical parity-check matrix yields bounded-degree, geometrically nonlocal, noncommuting targets with d_ord=3 and d_rel=Θ(m). To our knowledge, this gives a new class of efficiently preparable TFDs.
ER -