Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, Fernando G. S. L. Brandão
This chapter covers the quantum algorithmic primitive called Gibbs sampling. Gibbs sampling accomplishes the task of preparing a digital representation of the thermal state, also known as the Gibbs state, of a quantum system in thermal equilibrium. Gibbs sampling is an important ingredient in quantum algorithms to simulate physical systems. We cover multiple approaches to Gibbs sampling, including algorithms that are analogues of classical Markov chain Monte Carlo algorithms.
This chapter covers the quantum algorithmic primitive called Gibbs sampling. Gibbs sampling accomplishes the task of preparing a digital representation of the thermal state, also known as the Gibbs state, of a quantum system in thermal equilibrium. Gibbs sampling is an important ingredient in quantum algorithms to simulate physical systems. We cover multiple approaches to Gibbs sampling, including algorithms that are analogues of classical Markov chain Monte Carlo algorithms.
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.
Computations of chemical systems' equilibrium properties and non-equilibrium dynamics have been suspected of being a "killer app" for quantum computers. This review highlights the recent advancements of quantum algorithms tackling complex sampling tasks in the key areas of computational chemistry: ground state, thermal state properties, and quantum dynamics calculations. We review a broad range of quantum algorithms, from hybrid quantum-classical to fully quantum, focusing on the traditional Monte Carlo family, including Markov chain Monte Carlo, variational Monte Carlo, projector Monte Carlo, path integral Monte Carlo, etc. We also cover other relevant techniques involving complex sampling tasks, such as quantum-selected configuration interaction, minimally entangled typical thermal states, entanglement forging, and Monte Carlo-flavored Lindbladian dynamics. We provide a comprehensive overview of these algorithms' classical and quantum counterparts, detailing their theoretical frameworks and discussing the potentials and challenges in achieving quantum computational advantages.
Over the past decades, the problem of classical Gibbs sampling has led to one of the most powerful algorithmic families of the computer age: the Markov Chain Monte Carlo algorithms. Recent theoretical advances in quantum Gibbs sampling have, understandably, sparked considerable excitement. A recent breakthrough provided the first Lindbladians that provably drive a general quantum system to its Gibbs state, efficiently up to the mixing time. In this thesis, we bridge the gap between theory and implementation, with both analytical and numerical contributions. We work out every detail -- from transition rate choices, through quadrature, Trotter, and generator-splitting errors, to quantum circuits that could run on early fault-tolerant quantum computers. Using our fast Krylov-based Lindblad simulators and analysers, we numerically confirm existing theoretical Gibbs sampling results, cross-check our theoretical contributions, and do extrapolatory simulations to understand how the structure of these Gibbs samplers affects convergence rates. Our open-source framework, QuantumFurnace.jl, is built to work hand in hand with theory, providing the community with a tool to further advance quantum Gibbs sampling.
Sampling from thermal equilibrium distributions of quantum systems is a fundamental problem in multiple areas of science, such as condensed matter physics, quantum chemistry, and high-energy physics. Classical methods can degrade quickly, due to the fermionic sign problem or classical approximation errors, motivating the search for more efficient alternatives on quantum hardware. The presented work develops the Quantum Metropolis algorithm proposed by Temme et al. (2011) further to arbitrary energy precision while also reducing demands on quantum resources by requiring only a single, non-persistent auxiliary register. A general formalism for quantum Markov Chain Monte Carlo (qMCMC) maps is introduced and used to prove correctness via primitivity and quantum detailed balance. The simplified rejection mechanism results in additional conditions on the step operators, in particular requiring them to be permutations on the eigenbasis. Numerical simulations validate convergence behavior and robustness under ideal and approximate step operators. However, since constructing step operators directly based on the eigenbasis is not feasible in general, future work could explore variational learning of approximate step operators, structured Hamiltonians, or hybrid classical-quantum walk sampling.
Efficient sampling from a classical Gibbs distribution is an important computational problem with applications ranging from statistical physics over Monte Carlo and optimization algorithms to machine learning. We introduce a family of quantum algorithms that provide unbiased samples by preparing a state encoding the entire Gibbs distribution. We show that this approach leads to a speedup over a classical Markov chain algorithm for several examples, including the Ising model and sampling from weighted independent sets of two different graphs. Our approach connects computational complexity with phase transitions, providing a physical interpretation of quantum speedup. Moreover, it opens the door to exploring potentially useful sampling algorithms on near-term quantum devices, as the algorithm for sampling from independent sets on certain graphs can be naturally implemented using Rydberg atom arrays.