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arXiv:2610.06764

Hypercontractivity and entropy decay for infinite-dimensional quantum Gibbs samplers

Simon Becker, Cambyse Rouzé, Robert Salzmann

quant-phmath-phSubmitted Oct 5, 2026• Announced Oct 6, 2026• 0 citations
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Abstract

We derive rates of exponential relative-entropy decay for a broad class of infinite-dimensional quantum Gibbs samplers. A key technical ingredient is a perturbative stability theory for hypercontractive estimates and logarithmic Sobolev inequalities. For suitably chosen dissipative couplings, we obtain exponential entropy decay for full multimode Bose-Hubbard samplers with arbitrary hopping and repulsive density interactions. For harmonically confined quantum Coulomb gases and molecular systems, we prove exponential entropy decay for every fixed finite-rank low-energy truncation of the interaction. Finally, we exhibit a periodic charging model with a positive spectral gap but no finite-time $L^2$-to-$L^p$ bound for any $p>2$.

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We derive rates of exponential relative-entropy decay for a broad class of infinite-dimensional quantum Gibbs samplers. A key technical ingredient is a perturbative stability theory for hypercontractive estimates and logarithmic Sobolev inequalities. For suitably chosen dissipative couplings, we obtain exponential entropy decay for full multimode Bose-Hubbard samplers with arbitrary hopping and repulsive density interactions. For harmonically confined quantum Coulomb gases and molecular systems, we prove exponential entropy decay for every fixed finite-rank low-energy truncation of the interaction. Finally, we exhibit a periodic charging model with a positive spectral gap but no finite-time L2L^2L2-to-LpL^pLp bound for any p>2p>2p>2.
We propose a new method for proving L2L^2L2 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 L2L^2L2 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.
We introduce the notion of the BKM coercivity constant and show that the positivity of this constant governs the exponential relative entropy decay of the semigroup. Indeed, we prove that the modified log-Sobolev constant is equivalent to the BKM coercivity constant up to a constant depending on the asymptotic conditional expectation. We also discover that the BKM coercivity constants for matrix amplifications are already attained with a qubit auxiliary system, which also coincides with the GNS spectral gap of the generating Lindbladian. As a corollary, we establish a criterion that the complete modified log-Sobolev inequality holds if and only if the GNS gap of the semigroup is strictly positive. All the discussion above applies to quantum Markov semigroups that admit a faithful asymptotic conditional expectation as long-time equilibration, with no detailed balance condition assumed. As an application, we show the finite-dimensional Chen-Kastoryano-Gilyén Gibbs samplers always satisfy the complete modified log-Sobolev inequality.
We prove rapid mixing of a quantum Gibbs sampler for general one-dimensional non-commuting local spin systems at every fixed positive temperature. In particular, for a system of nnn sites, we show that the Kubo--Martin--Schwinger (KMS) detailed-balanced Gibbs sampler of~\cite{CKG2023} converges from any initial state to trace-norm error εεε in time O(log⁡4(n)+log⁡(1/ε))O(\log^4(n)+\log(1/ε))O(log4(n)+log(1/ε)). This improves the previous O(n+log⁡(1/ε))O(n+\log(1/ε))O(n+log(1/ε)) bound and establishes rapid mixing beyond high-temperature and weak-interaction regimes. Our proof explains why the slowest decay rate, captured by the spectral gap, can substantially underestimate the speed of equilibration. The key insight is that a state far from equilibrium must have a substantial component outside the span of low-weight Pauli operators, where stronger dissipation estimates apply. This yields two stages of relaxation: a double-exponential convergence from a bad initial state, followed by exponential convergence near equilibrium. We turn this mechanism into a framework for proving rapid mixing through spectral analysis beyond the gap. Our decay estimate is equivalent to exponential convergence in sandwiched Rényi-2 divergence. Together with the spectral gap, it also implies a modified logarithmic Sobolev inequality with rate Ω(log⁡−4(n))Ω(\log^{-4}(n))Ω(log−4(n)).
We develop a quantum Gibbs-sampling approach to free-energy estimation for trapped systems of distinguishable particles with Coulomb interactions in dimensions d∈{2,3}d\in\{2,3\}d∈{2,3}. The singular infinite-dimensional problem is reduced to a finite-rank low-energy truncation of the interaction, the truncated Gibbs state is prepared by a quantum Markov semigroup, and thermodynamic integration gives the free energy. We obtain explicit truncation rates, quantitative spectral-gap estimates, and a finite-dimensional circuit implementation. In particular, every fixed interaction truncation has a positive gap, while weak truncated interactions admit a particle-number-independent lower bound. The circuit complexity is conditional on a certified gap lower bound at the accuracy-dependent interaction cutoff.
We develop a rigorous and implementable framework for Gibbs sampling of infinite-dimensional quantum systems governed by unbounded Hamiltonians. Extending dissipative Gibbs samplers beyond finite dimensions raises fundamental obstacles, including ill-defined generators, the absence of spectral gaps on natural Banach spaces, and tensions between implementability and convergence guarantees. We overcome these issues by constructing KMS-symmetric quantum Markov semigroups on separable Hilbert spaces that are both well-posed and efficiently implementable on qubit hardware. Our generation theory is based on the abstract framework of Dirichlet forms, adapted here to the case of algebras of bounded operators over separable Hilbert spaces. Leveraging the spectral properties of our self-adjoint generators, we establish quantitative convergence results in trace distance, including regimes of fast thermalization. In contrast, we also identify Hamiltonians for which a naive choice of generators guaranteeing implementability generally comes at the cost of losing uniform exponential convergence guarantees for the associated evolutions, thereby establishing a strong trade-off between implementability and convergence. Our framework applies to a wide class of models - including Schrödinger operators, Gaussian systems, and Bose-Hubbard Hamiltonians - and provides a unified approach linking rigorous infinite-dimensional analysis with algorithmic Gibbs state preparation.
Hypercontractivity of a quantum dynamical semigroup has strong implications for its convergence behavior and entropy decay rate. A logarithmic Sobolev inequality and the corresponding logarithmic Sobolev constant can be inferred from the semigroup's hypercontractive norm bound. We consider completely-positive quantum mechanical semigroups described by a Lindblad master equation. To prove the norm bound, we follow an approach which has its roots in the study of classical rate equations. We use interpolation theorems for non-commutative LpL_pLp​ spaces to obtain a general hypercontractive inequality from a particular p→qp \rightarrow qp→q-norm bound. Then, we derive a bound on the 2→42 \rightarrow 42→4-norm from an analysis of the block diagonal structure of the semigroup's spectrum. We show that the dynamics of an NNN-qubit graph state Hamiltonian weakly coupled to a thermal environment is hypercontractive. As a consequence this allows for the efficient preparation of graph states in time poly(log⁡(N)){\rm poly}(\log(N))poly(log(N)) by coupling at sufficiently low temperature. Furthermore, we extend our results to gapped Liouvillians arising from a weak linear coupling of a free-fermion systems.