aiqbench analysis
This paper gives a general method for proving logarithmic Sobolev inequalities whose constants don't grow with system size, for quantum Gibbs samplers that satisfy KMS detailed balance. The approach transports the inequality from the infinite-temperature product state to the target Gibbs state, then uses a tensor-power argument with the quantum asymptotic equipartition property to reach relative entropy. For quantum computing readers, the practical relevance is that such bounds underpin rapid mixing, and so efficient preparation of thermal states, for 1D spin chains at any finite temperature and for high-temperature finite-range systems in any dimension. It is theoretical work, so its results are not yet tied to hardware or algorithm performance.
Size-independent Logarithmic Sobolev Inequalities for Quantum Gibbs Samplers
arXiv:2610.31571v1 Announce Type: new Abstract: We propose a new method for proving $L^2$ and modified logarithmic Sobolev inequalities with constants independent of system size for quantum Gibbs samplers whose Lindblad generators satisfy Kubo-Martin-Schwinger (KMS) detailed balance. We first establish the $L^2$ logarithmic Sobolev inequality by transporting it along a temperature-dependent path from the infinite-temperature product state to the target Gibbs state while maintaining a size-independent decay rate for the sandwiched R\'enyi-2 divergence. A tensor-power argument, combined with the quantum asymptotic equipartition property, then transfers this decay bound to relative entropy, yielding the modified logarithmic Sobolev inequality. This approach provides a general framework for proving size-independent logarithmic Sobolev inequalities for quantum Gibbs samplers. We apply it to two regimes: one-dimensional quantum spin chains at every finite temperature, and finite-range quantum spin systems at sufficiently high temperature in any fixed dimension.
Abstract:We propose a new method for proving $L^2$ and modified logarithmic Sobolev inequalities with constants independent of system size for quantum Gibbs samplers whose Lindblad generators satisfy Kubo-Martin-Schwinger (KMS) detailed balance. We first establish the $L^2$ logarithmic Sobolev inequality by transporting it along a temperature-dependent path from the infinite-temperature product state to the target Gibbs state while maintaining a size-independent decay rate for the sandwiched Rényi-2 divergence. A tensor-power argument, combined with the quantum asymptotic equipartition property, then transfers this decay bound to relative entropy, yielding the modified logarithmic Sobolev inequality. This approach provides a general framework for proving size-independent logarithmic Sobolev inequalities for quantum Gibbs samplers. We apply it to two regimes: one-dimensional quantum spin chains at every finite temperature, and finite-range quantum spin systems at sufficiently high temperature in any fixed dimension.
arXiv:2610.31571 [quant-ph]
(or arXiv:2610.31571v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2610.31571 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 225 words