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.
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.
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.
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 β, up to polylogarithmic factors. Moreover, the gate complexity reduces significantly for lattice Hamiltonians as the corresponding Lindblad operators are (quasi-) local (with radius ∼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.
Preparing ground states and thermal states is essential for simulating quantum systems on quantum computers. Despite the hope for practical quantum advantage in quantum simulation, popular state preparation approaches have been challenged. Monte Carlo-style quantum Gibbs samplers have emerged as an alternative, but prior proposals have been unsatisfactory due to technical obstacles rooted in energy-time uncertainty. We introduce simple continuous-time quantum Gibbs samplers that overcome these obstacles by efficiently simulating Nature-inspired quantum master equations (Lindbladians). In addition, we construct the first provably accurate and efficient algorithm for preparing certain purified Gibbs states (called thermal field double states in high-energy physics) of rapidly thermalizing systems; this algorithm also benefits from a quantum walk speedup. Our algorithms' costs have a provable dependence on temperature, accuracy, and the mixing time (or spectral gap) of the relevant Lindbladian. We complete the first rigorous proof of finite-time thermalization for physically derived Lindbladians by developing a general analytic framework for nonasymptotic secular approximation and approximate detailed balance. Given the success of classical Markov chain Monte Carlo (MCMC) algorithms and the ubiquity of thermodynamics, we anticipate that quantum Gibbs sampling will become indispensable in quantum computing.
Systems in thermal equilibrium at non-zero temperature are described by their Gibbs state. For classical many-body systems, the Metropolis-Hastings algorithm gives a Markov process with a local update rule that samples from the Gibbs distribution. For quantum systems, sampling from the Gibbs state is significantly more challenging. Many algorithms have been proposed, but these are more complex than the simple local update rule of classical Metropolis sampling, requiring non-trivial quantum algorithms such as phase estimation as a subroutine. Here, we show that a dissipative quantum algorithm with a simple, local update rule is able to sample from the quantum Gibbs state. In contrast to the classical case, the quantum Gibbs state is not generated by converging to the fixed point of a Markov process, but by the states generated at the stopping time of a conditionally stopped process. This gives a new answer to the long-sought-after quantum analogue of Metropolis sampling. Compared to previous quantum Gibbs sampling algorithms, the local update rule of the process has a simple implementation, which may make it more amenable to near-term implementation on suitable quantum hardware. This dissipative Gibbs sampler works for arbitrary quantum Hamiltonians, without any assumptions on or knowledge of its properties, and comes with certifiable precision and run-time bounds. We also show that the algorithm benefits from some measure of built-in resilience to faults and errors (``fault resilience''). Finally, we also demonstrate how the stopping statistics of an ensemble of runs of the dissipative Gibbs sampler can be used to estimate the partition function.
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.
Gibbs sampling is a crucial computational technique used in physics, statistics, and many other scientific fields. For classical Hamiltonians, the most commonly used Gibbs sampler is the Metropolis algorithm, known for having the Gibbs state as its unique fixed point. For quantum Hamiltonians, designing provably correct Gibbs samplers has been more challenging. [TOV+11] introduced a novel method that uses quantum phase estimation (QPE) and the Marriot-Watrous rewinding technique to mimic the classical Metropolis algorithm for quantum Hamiltonians. The analysis of their algorithm relies upon the use of a boosted and shift-invariant version of QPE which may not exist [CKBG23]. Recent efforts to design quantum Gibbs samplers take a very different approach and are based on simulating Davies generators [CKBG23,CKG23,RWW23,DLL24]. Currently, these are the only provably correct Gibbs samplers for quantum Hamiltonians. We revisit the inspiration for the Metropolis-style algorithm of [TOV+11] and incorporate weak measurement to design a conceptually simple and provably correct quantum Gibbs sampler, with the Gibbs state as its approximate unique fixed point. Our method uses a Boosted QPE which takes the median of multiple runs of QPE, but we do not require the shift-invariant property. In addition, we do not use the Marriott-Watrous rewinding technique which simplifies the algorithm significantly.