Adiabatic tracking and geometric schedules for classical and quantum Gibbs-state preparation
Year: 2026
Type: preprint
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 $\dotβ\proptoδ(β)/V(β)$, where $V(β)=\sqrt{\operatorname{Var}_{π_β}(E)}$. We then apply the discrete quantum adiabatic theorem to temperature-dependent Szegedy quantum walks, whose eigenphase gap satisfies $Δ_w=Θ(\sqrt{δ_+})$, 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.
Source: arXiv (Cornell University)
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220706669