Dissipative Quantum Gibbs Sampling

TL;DR

This work introduces the dissipative Gibbs sampler (DGS), a local-update quantum algorithm that samples from the Gibbs state $\rho_G(H)$ by a stopping-time rule rather than fixed-point convergence, addressing the challenge of quantum Gibbs sampling. It provides a precise construction using a local instrument and a weak-measurement based Kraus operator $K$, together with carefully chosen stopping probabilities, to achieve provable approximation to $\rho_G(H)$ with explicit run-time bounds. The paper also shows how to estimate the partition function from stopping statistics and extends the approach to prepare more general functions of the Hamiltonian, while proving fault resilience under a bounded noise model. These results, grounded in the dissipative quantum eigensolver framework, suggest a practical, hardware-friendly path to quantum Gibbs sampling and observables on near-term devices, including potential parallelization and partition-function estimation capabilities.

Abstract

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.

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Theorems & Definitions (24)

• Theorem 2
• Lemma 3
• proof
• Corollary 4
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• Lemma 6
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• Lemma 7
• proof