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arXiv:2404.05998 · Communications in Mathematical Physics

Efficient Quantum Gibbs Samplers with Kubo–Martin–Schwinger Detailed Balance Condition

Zhiyan Ding, Bowen Li, Lin Lin

Published Feb 27, 2025• 35 citations
arXivHTMLPDF

Abstract

Abstract Lindblad dynamics and other open-system dynamics provide a promising path towards efficient Gibbs sampling on quantum computers. In these proposals, the Lindbladian is obtained via an algorithmic construction akin to designing an artificial thermostat in classical Monte Carlo or molecular dynamics methods, rather than being treated as an approximation to weakly coupled system-bath unitary dynamics. Recently, Chen, Kastoryano, and Gilyén (arXiv:2311.09207) introduced the first efficiently implementable Lindbladian satisfying the Kubo–Martin–Schwinger (KMS) detailed balance condition, which ensures that the Gibbs state is a fixed point of the dynamics and is applicable to non-commuting Hamiltonians. This Gibbs sampler uses a continuously parameterized set of jump operators, and the energy resolution required for implementing each jump operator depends only logarithmically on the precision and the mixing time. In this work, we build upon the structural characterization of KMS detailed balanced Lindbladians by Fagnola and Umanità, and develop a family of efficient quantum Gibbs samplers using a finite set of jump operators (the number can be as few as one), akin to the classical Markov chain-based sampling algorithm. Compared to the existing works, our quantum Gibbs samplers have a comparable quantum simulation cost but with greater design flexibility and a much simpler implementation and error analysis. Moreover, it encompasses the construction of Chen, Kastoryano, and Gilyén as a special instance.

DOI
10.1007/s00220-025-05235-3

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Abstract Lindblad dynamics and other open-system dynamics provide a promising path towards efficient Gibbs sampling on quantum computers. In these proposals, the Lindbladian is obtained via an algorithmic construction akin to designing an artificial thermostat in classical Monte Carlo or molecular dynamics methods, rather than being treated as an approximation to weakly coupled system-bath unitary dynamics. Recently, Chen, Kastoryano, and Gilyén (arXiv:2311.09207) introduced the first efficiently implementable Lindbladian satisfying the Kubo–Martin–Schwinger (KMS) detailed balance condition, which ensures that the Gibbs state is a fixed point of the dynamics and is applicable to non-commuting Hamiltonians. This Gibbs sampler uses a continuously parameterized set of jump operators, and the energy resolution required for implementing each jump operator depends only logarithmically on the precision and the mixing time. In this work, we build upon the structural characterization of KMS detailed balanced Lindbladians by Fagnola and Umanità, and develop a family of efficient quantum Gibbs samplers using a finite set of jump operators (the number can be as few as one), akin to the classical Markov chain-based sampling algorithm. Compared to the existing works, our quantum Gibbs samplers have a comparable quantum simulation cost but with greater design flexibility and a much simpler implementation and error analysis. Moreover, it encompasses the construction of Chen, Kastoryano, and Gilyén as a special instance.
Classical Markov Chain Monte Carlo methods have been essential for simulating statistical physical systems and have proven well applicable to other systems with many degrees of freedom. Motivated by the statistical physics origins, Chen, Kastoryano, and Gilyén [CKG23] proposed a continuous-time quantum thermodynamic analogue to Glauber dynamics that is (i) exactly detailed balanced, (ii) efficiently implementable, and (iii) quasi-local for geometrically local systems. Physically, their construction resembles the dissipative dynamics arising from weak system-bath interaction. In this work, we give an efficiently implementable discrete-time counterpart to any continuous-time quantum Gibbs sampler. Our construction preserves the desirable features (i)-(iii) while does not decrease the spectral gap. Also, we give an alternative highly coherent quantum generalization of detailed balanced dynamics that resembles another physically derived master equation, and propose a smooth interpolation between this and earlier constructions. Moreover, we show how to make earlier Metropolis-style Gibbs samplers (which estimate energies both before and after jumps) exactly detailed balanced. We study generic properties of all constructions, including the uniqueness of the fixed point, the (quasi-)locality of the resulting operators. Finally, we prove that the spectral gap of our new highly coherent Gibbs sampler is constant at high temperatures, thereby it mixes fast. We hope that our systematic approach to quantum Glauber and Metropolis dynamics will lead to widespread applications in various domains.
Szegedy's quantum walk quadratically accelerates reversible classical Markov chains, but extending this mechanism to efficiently implementable quantum Gibbs samplers has remained challenging beyond special classes. We present a walk-free quantum algorithm for preparing purified Gibbs states with a quadratic improvement in spectral-gap dependence for a broad class of quantum Gibbs samplers that satisfy exact Kubo-Martin-Schwinger detailed balance. Our main structural result is an explicit factorization of the corresponding parent Hamiltonian into noncommutative first-order operators. This turns purified Gibbs-state preparation into a singular-value filtering problem and enables a quantum singular value transformation algorithm with quadratically improved gap dependence under standard coherent-access and warm-start assumptions. The framework applies to several efficiently implementable Gibbs samplers beyond the Davies setting. Moreover, we construct noncommuting Hamiltonians with efficiently preparable warm starts whose refinement under quantum Gibbs samplers requires exponential time. For these models, our algorithm achieves an end-to-end quantum speedup over the worst-case mixing scale.
Preparing thermal and ground states is an essential quantum algorithmic task for quantum simulation. In this work, we construct the first efficiently implementable and exactly detailed-balanced Lindbladian for Gibbs states of arbitrary noncommutative Hamiltonians. Our construction can also be regarded as a continuous-time quantum analog of the Metropolis-Hastings algorithm. To prepare the quantum Gibbs state, our algorithm invokes Hamiltonian simulation for a time proportional to the mixing time and the inverse temperature βββ, up to polylogarithmic factors. Moreover, the gate complexity reduces significantly for lattice Hamiltonians as the corresponding Lindblad operators are (quasi-) local (with radius ∼β\simβ∼β) and only depend on local Hamiltonian patches. Meanwhile, purifying our Lindbladians yields a temperature-dependent family of frustration-free "parent Hamiltonians", prescribing an adiabatic path for the canonical purified Gibbs state (i.e., the Thermal Field Double state). These favorable features suggest that our construction serves as a quantum algorithmic counterpart to classical Markov chain Monte Carlo sampling.
We investigate quantum thermal state preparation algorithms based on system-bath interactions and uncover a surprising phenomenon in the weak-coupling regime. We rigorously prove that, if the system-bath interaction is engineered so that the transition part of the approximate Lindbladian generator satisfies the Kubo--Martin--Schwinger (KMS) detailed balance condition, then the unique fixed point of the dynamics can be made arbitrarily close to the Gibbs state in the weak-coupling limit, regardless of the structure of the Lamb shift term. Importantly, this remains true even when the approximate Lindbladian differs substantially from the ideal Davies generator and the Lamb shift term does not commute with the thermal state. Our result shows that the role of the KMS detailed balance condition extends well beyond standard Lindbladian dynamics, serving as a general principle for a broader class of dissipative systems. Furthermore, by combining this with a general perturbation framework, we bound the mixing time of the dynamics and establish an end-to-end complexity of O(ε−1)\mathcal{O}(\varepsilon^{-1})O(ε−1) for Gibbs state preparation. These guarantees apply to any Hamiltonian whose associated KMS-detailed-balance Lindbladian is known to be fast mixing.
Understanding when quantum many-body systems rapidly converge to thermal equilibrium is a central problem in quantum statistical mechanics. While spatial locality provides powerful tools for systems with an underlying geometry, fully connected models require estimates that accommodate interactions between every pair of spins. We study Gibbs-state preparation for two-local quantum Hamiltonians on arbitrary interaction graphs, including fully connected systems, via Lindblad dynamics that satisfies Kubo--Martin--Schwinger (KMS) detailed balance. Our approach is based on a quantum analogue of the classical Dobrushin condition, which controls the influence of local perturbations on the rest of the system. Using the quantum Wasserstein distance of order 111, we quantify how these perturbations propagate under the dynamics and derive contraction estimates in a high-temperature regime controlled by the maximum total interaction strength per site. These estimates require no assumptions on spatial geometry or graph growth and yield mixing times logarithmic in the system size and the inverse error tolerance, uniformly over all initial states.
Recent progress in fully quantum Markov chain Monte Carlo methods enables efficient Gibbs-state sampling on quantum computers [Chen et al., Nature 646, 561 (2025)]. Although rigorous worst-case bounds on mixing times remain largely inaccessible for classically intractable systems, experience from classical Monte Carlo suggests that convergence of relevant observables may nevertheless be rapid. This raises the practical question of how to diagnose convergence efficiently, i.e., with at most polynomial overhead. We propose a low-cost criterion for convergence monitoring that exploits the weak measurements inherent in quantum Gibbs samplers and their qubit-efficient variants [Ding et al., arXiv:2508.05703 (2025)]. Our approach is based on the observation that, at thermal equilibrium, the net energy flow between system and environment vanishes and energy-exchange statistics satisfy a balance condition. This condition appears in the distribution of (quasi-)frequencies extracted from the weak-measurement record and we use it to construct a Hamiltonian-agnostic stopping criterion based solely on data already generated by the sampler. We provide a statistical analysis, along with numerical and analytical studies to understand its performance, assumptions, and limitations.