Adiabatic tracking and geometric schedules for classical and quantum Gibbs-state preparation
Abstract
We develop a discrete adiabatic framework for Gibbs-state preparation in classical and quantum optimisation algorithms. For finite time-inhomogeneous reversible Markov chains, we derive a tracking theorem that bounds the distance between the evolving distribution and the instantaneous stationary state. The bound combines contraction of accumulated error with the new error generated as the stationary distribution changes. For Gibbs paths, it yields explicit step-complexity bounds and an effective tracking metric determined by the spectral gap and equilibrium energy fluctuations. The associated constant-speed schedule minimizes the metric-energy functional and, in the small-gap regime, takes the form β̇∝δ(β)/V(β), where V(β)=√(Varπβ(E)). We then apply the discrete quantum adiabatic theorem to temperature-dependent Szegedy quantum walks, whose eigenphase gap satisfies Δw=Θ(√(δ+)), with δ+ the underlying Markov kernel's right spectral gap. Under uniform bounds on the discrete operator variation and suitable spectral and path-integral assumptions, gap-adapted schedules give a sufficient fixed-accuracy cost of O(δ+,min−1/2). This gives a quadratic improvement in gap dependence relative to the classical inverse-gap relaxation scale under the stated assumptions. Numerical tests on a solvable two-state model and Sherrington-Kirkpatrick instances compare tracking and final preparation errors and investigate the transfer of schedules from smaller to larger systems.
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BibTeX
@misc{costa2026adiabatic,
title = {{Adiabatic tracking and geometric schedules for classical and quantum Gibbs-state preparation}},
author = {Pedro C. S. Costa and Mauro E. S. Morales and Dong An and Kaur Kristjuhan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06268},
eprint = {2610.06268},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06268},
}
RIS
TY - UNPB
TI - Adiabatic tracking and geometric schedules for classical and quantum Gibbs-state preparation
AU - Pedro C. S. Costa
AU - Mauro E. S. Morales
AU - Dong An
AU - Kaur Kristjuhan
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.06268
UR - https://arxiv.org/abs/2610.06268
AB - We develop a discrete adiabatic framework for Gibbs-state preparation in classical and quantum optimisation algorithms. For finite time-inhomogeneous reversible Markov chains, we derive a tracking theorem that bounds the distance between the evolving distribution and the instantaneous stationary state. The bound combines contraction of accumulated error with the new error generated as the stationary distribution changes. For Gibbs paths, it yields explicit step-complexity bounds and an effective tracking metric determined by the spectral gap and equilibrium energy fluctuations. The associated constant-speed schedule minimizes the metric-energy functional and, in the small-gap regime, takes the form β̇∝δ(β)/V(β), where V(β)=√(Var_π_β(E)). We then apply the discrete quantum adiabatic theorem to temperature-dependent Szegedy quantum walks, whose eigenphase gap satisfies Δ_w=Θ(√(δ_+)), with δ_+ the underlying Markov kernel's right spectral gap. Under uniform bounds on the discrete operator variation and suitable spectral and path-integral assumptions, gap-adapted schedules give a sufficient fixed-accuracy cost of O(δ_+,min^−1/2). This gives a quadratic improvement in gap dependence relative to the classical inverse-gap relaxation scale under the stated assumptions. Numerical tests on a solvable two-state model and Sherrington-Kirkpatrick instances compare tracking and final preparation errors and investigate the transfer of schedules from smaller to larger systems.
ER -