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arXiv:2609.36016

Is rapid mixing stable?

Jordi A. Montañà-López, Barbara Roos, Sebastian Stengele, Ángela Capel, Rahul Trivedi

quant-phmath-phSubmitted Sep 28, 2026• Announced Sep 30, 2026• 0 citations
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Abstract

Several physically interesting Lindbladians, such as those describing generic open quantum system models as well as engineered state preparation processes, are often rapidly mixing, that is, they relax to steady states in time scaling polylogarithmically with the system size. Owing to the short mixing times, it is physically expected that local perturbations do not substantially increase the mixing time. In this work we question this expectation and ask whether rapid mixing survives. We provide counterexamples showing that rapid mixing is not stable in general, even when the perturbation does not change the steady state. We also show that arbitrarily weak local Hamiltonian perturbations can destroy rapid mixing. Complementing these results, we develop an array of sufficient conditions which imply stable rapid mixing, including several classes of quasifree fermionic systems, commuting Lindbladians, and local spin systems. We also identify conditions under which modified logarithmic Sobolev inequalities (MLSIs), a stronger form of convergence, lead to stable rapid mixing. As a further implication of our results, we show that rapid mixing does not imply a positive MLSI constant.

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Several physically interesting Lindbladians, such as those describing generic open quantum system models as well as engineered state preparation processes, are often rapidly mixing, that is, they relax to steady states in time scaling polylogarithmically with the system size. Owing to the short mixing times, it is physically expected that local perturbations do not substantially increase the mixing time. In this work we question this expectation and ask whether rapid mixing survives. We provide counterexamples showing that rapid mixing is not stable in general, even when the perturbation does not change the steady state. We also show that arbitrarily weak local Hamiltonian perturbations can destroy rapid mixing. Complementing these results, we develop an array of sufficient conditions which imply stable rapid mixing, including several classes of quasifree fermionic systems, commuting Lindbladians, and local spin systems. We also identify conditions under which modified logarithmic Sobolev inequalities (MLSIs), a stronger form of convergence, lead to stable rapid mixing. As a further implication of our results, we show that rapid mixing does not imply a positive MLSI constant.
Dissipative quantum algorithms for state preparation in many-body systems are increasingly recognised as promising candidates for achieving large quantum advantages in application-relevant tasks. Recent advances in algorithmic, detailed-balance Lindbladians enable the efficient simulation of open-system dynamics converging towards desired target states. However, the overall complexity of such schemes is governed by system-size dependent mixing times. In this work, we analyse algorithmic Lindbladians for Gibbs state preparation and prove that they exhibit rapid mixing, i.e., convergence in time poly-logarithmic in the system size. We first establish this for non-interacting spin systems, free fermions, and free bosons, and then show that these rapid mixing results are stable under perturbations, covering weakly interacting qudits and perturbed non-hopping fermions. Further, we adapt the techniques from separable qudits to the fermionic setting and prove rapid mixing of the strongly-interacting regime of the Fermi-Hubbard model, for which we also explicitly evaluate the guaranteed parameter regimes. Our results constitute the first efficient mixing bounds for non-commuting qudit models and bosonic systems at arbitrary temperatures. Compared to prior spectral-gap-based results for fermions, we achieve exponentially faster mixing, further featuring explicit constants on the maximal allowed interaction strength. This not only improves the overall polynomial runtime for quantum Gibbs state preparation, but also enhances robustness against noise. Our analysis relies on oscillator norm techniques from mathematical physics, where we introduce tailored variants adapted to specific Lindbladians $\unicode{x2014}$ an innovation that we expect to significantly broaden the scope of these methods.
Markovian open-system dynamics have widespread applications throughout quantum information science, including algorithmic state preparation. Their convergence is commonly quantified using the worst case global trace distance between the evolving and stationary states. However, this criterion can be unnecessarily stringent when only physically relevant observables are of interest. Here we introduce and study observable-specific mixing times. We prove that, for quasi-local, rapidly mixing Lindbladians, sums of geometrically local observables equilibrate in a time independent of system size, in contrast to the logarithmic dependence of global state mixing. This separation reduces the runtime of dissipative quantum algorithms, including quantum Gibbs samplers, for estimating quantities such as the Gibbs state energy and local order parameters, yielding an overall scaling that is linear in system size. Complementing this quantum result, we develop a quantum-inspired classical algorithm for estimating the same quantities. Its runtime is likewise linear in system size, but scaling exponentially in O(log⁡(1/ε)D)\mathcal{O}\big(\log(1/ε)^D\big)O(log(1/ε)D), where DDD denotes the spatial dimension of the lattice. We further analyse non-interacting Lindbladians over qudits, fermions, and bosons, demonstrating that locality of observables is not always necessary for a qualitatively faster mixing. Small-scale simulations of quantum Gibbs samplers reveal no large hidden constants in our asymptotic analysis and show that the theoretical predictions closely capture the finite-size dynamics.
In this thesis we study properties of open quantum dissipative evolutions of spin systems on lattices described by Lindblad generators, in a particular regime that we denote rapid mixing. We consider dissipative evolutions with a unique fixed point, and which compress the whole space of input states into increasingly small neighborhoods of the fixed point. The time scale at which this compression takes place, or in other words the time we have to wait for any input state to become almost indistinguishable from the fixed point, is called the mixing time of the process. Rapid mixing is a condition on the scaling of this mixing time with the system size: if it is logarithmic, then we have rapid mixing. The main contribution of this thesis is to show that rapid mixing has profound implications for the corresponding system: it is stable against external perturbations and its fixed point satisfies an area law for mutual information.
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)).
Abstract Quantum systems typically reach thermal equilibrium rather quickly when coupled to a thermal environment. The usual way of bounding the speed of this process is by estimating the spectral gap of the dissipative generator. However the gap, by itself, does not always yield a reasonable estimate for the thermalization time in many-body systems: without further structure, a uniform lower bound on it only constrains the thermalization time to grow polynomially with system size. Here, instead, we show that for a large class of geometrically-2-local models of Davies generators with commuting Hamiltonians, the thermalization time is much shorter than one would naïvely estimate from the gap: at most logarithmic in the system size. This yields the so-called rapid mixing of dissipative dynamics. The result is particularly relevant for 1D systems, for which we prove rapid thermalization with a system size independent decay rate only from a positive gap in the generator. We also prove that systems in hypercubic lattices of any dimension, and exponential graphs, such as trees, have rapid mixing at high enough temperatures. We do this by introducing a novel notion of clustering which we call “strong local indistinguishability” based on a max-relative entropy, and then proving that it implies a lower bound on the modified logarithmic Sobolev inequality (MLSI) for nearest neighbour commuting models. This has consequences for the rate of thermalization towards Gibbs states, and also for their relevant Wasserstein distances and transportation cost inequalities. Along the way, we show that several measures of decay of correlations on Gibbs states of commuting Hamiltonians are equivalent, a result of independent interest. At the technical level, we also show a direct relation between properties of Davies and Schmidt dynamics, that allows to transfer results of thermalization between both.
We analyze the mixing time of open quantum systems governed by the Lindblad master equation, showing that it is determined not only by the Liouvillian gap, but also by the trace-norm factor of each decaying Liouvillian eigenmode. By utilizing them as universal predictors of mixing time, we establish general conditions for the fast and rapid mixing, respectively. Specifically, we derive rapid mixing conditions for both the strong and weak dissipation regimes, formulated as sparsity constraints on the Hamiltonian and the local Lindblad operators. Our findings provide a general framework for calculating mixing time and offer a guide for designing dissipation to achieve desired mixing speeds, which has significant implications for efficient experimental state preparation.