We establish a systematic framework of unbiased quantum sampling and estimation protocols for the classical Gibbs expectation. This framework generalizes existing approaches to the partition function estimation and has broader applications in various fields. We consider sampling and estimation for a wide class of non-log-concave distributions, particularly heavy-tailed ones, under relaxed assumptions beyond strong convexity, such as dissipativity.
We develop an unbiased extension of quantum-accelerated multilevel Monte Carlo (QA-MLMC) to eliminate all biases from discretization and time truncation, together with introducing a change-of-measure approach and the Girsanov theorem via Radon-Nikodym derivatives. As a result, our approach achieves quantum complexity $\widetilde{\mathcal{O}}(ε^{-1})$ within error $ε$, whereas the classical MLMC requires $\widetilde{\mathcal{O}}(ε^{-2})$ and existing quantum algorithms yield biased estimators under stronger assumptions. Furthermore, our unified framework enables unbiased quantum sampling and estimation for certain heavy-tailed distributions after transformation. We provide several concrete applications of our approach in statistics, machine learning, and finance, towards more practical scenarios of the quantum acceleration of stochastic processes.
We establish a systematic framework of unbiased quantum sampling and estimation protocols for the classical Gibbs expectation. This framework generalizes existing approaches to the partition function estimation and has broader applications in various fields. We consider sampling and estimation for a wide class of non-log-concave distributions, particularly heavy-tailed ones, under relaxed assumptions beyond strong convexity, such as dissipativity.
We develop an unbiased extension of quantum-accelerated multilevel Monte Carlo (QA-MLMC) to eliminate all biases from discretization and time truncation, together with introducing a change-of-measure approach and the Girsanov theorem via Radon-Nikodym derivatives. As a result, our approach achieves quantum complexity O(ε−1) within error ε, whereas the classical MLMC requires O(ε−2) and existing quantum algorithms yield biased estimators under stronger assumptions. Furthermore, our unified framework enables unbiased quantum sampling and estimation for certain heavy-tailed distributions after transformation. We provide several concrete applications of our approach in statistics, machine learning, and finance, towards more practical scenarios of the quantum acceleration of stochastic processes.
Given a convex function f:Rd→R, the problem of sampling from a distribution ∝e−f(x) is called log-concave sampling. This task has wide applications in machine learning, physics, statistics, etc. In this work, we develop quantum algorithms for sampling log-concave distributions and for estimating their normalizing constants ∫Rde−f(x)dx. First, we use underdamped Langevin diffusion to develop quantum algorithms that match the query complexity (in terms of the condition number κ and dimension d) of analogous classical algorithms that use gradient (first-order) queries, even though the quantum algorithms use only evaluation (zeroth-order) queries. For estimating normalizing constants, these algorithms also achieve quadratic speedup in the multiplicative error ε. Second, we develop quantum Metropolis-adjusted Langevin algorithms with query complexity O(κ1/2d) and O(κ1/2d3/2/ε) for log-concave sampling and normalizing constant estimation, respectively, achieving polynomial speedups in κ,d,ε over the best known classical algorithms by exploiting quantum analogs of the Monte Carlo method and quantum walks. We also prove a 1/ε1−o(1) quantum lower bound for estimating normalizing constants, implying near-optimality of our quantum algorithms in ε.
For a convex function f:Rd→R, the problem of sampling from a distribution proportional to e−f(x) is called log-concave sampling. In many practical scenarios, the function f(x) turns out to admit a local decomposition f(x)=∑a=1Rψa(xSa). In this paper, we consider log-concave sampling using local queries, i.e., evaluation and gradient queries to each clause ψa(⋅), which can be computationally much cheaper than the queries to f(x) itself. We show that if each coordinate appears in only a small number of clauses, there is a quantum algorithm for strongly log-concave sampling using O(κd) local queries, where κ is the condition number. This improves the prior best classical result O(κd) due to Ascolani, Lavenant, and Zanella (Ann. Probab. 2026) and the quantum result O(κd2) implied by Childs et al. (NeurIPS 2022). Our quantum sampler applies to a broad class of locally structured models from statistical computing and machine learning, with representative examples including Gaussian Markov random fields, finite-element latent Gaussian models, and sparse generalized linear models. These results demonstrate that local structure is not merely an implementation detail, but a quantum algorithmic resource for high-dimensional sampling.
We study stochastic optimization with heavy-tailed gradient noise. We first propose a novel quantum mean estimator for multivariate heavy-tailed random variables that achieves lower query complexity than optimal classical estimators in the low-dimensional regime. We further develop an unbiased quantum mean estimator by applying a generalized multi-level Monte Carlo technique. We prove quantum lower bounds showing that, when the dimension d of the random vector is small and can be viewed as a constant, our quantum estimators are optimal up to logarithmic factors. We further derive stronger dimension-dependent lower bounds for tail index p>4/3, showing that a nontrivial dependence on the dimension is unavoidable in the low-dimensional regime. Based on these estimators, we propose a quantum normalized stochastic gradient descent method (QNSGD), which finds an ε-stationary point using O~(dε−2p−25p−4) queries to the quantum stochastic gradient oracle. For a convex objective function, we propose a quantum projected stochastic gradient descent method (QPSGD), which computes a solution with ε-optimal solution using O~(dε−2p−23p−2+ε−2) queries in expectation. These sharper bounds improve upon the classical lower bounds Ω(ε−p−13p−2) for nonconvex problems and Ω(ε−p−1p) for convex problems in the low-dimensional regimes d≲ε−p−1p and d≲ε−p−12−p, respectively.
We present quantum speedups for sampling from distributions of the form π∝e−f on Rd. We consider two stochastic oracle models: a stochastic gradient oracle, where $f=\frac{1}{n}\sum_{i=1}^n f_i $ and component gradients {∇fi}i∈[n] are available, and a stochastic evaluation oracle, where only noisy values of f are available. Our framework accelerates classical stochastic Langevin Monte Carlo (LMC) and Hamiltonian Monte Carlo (HMC) algorithms by replacing stochastic gradient estimators with variance-controlled quantum mean estimation and gradient estimation subroutines. Unlike quantum walk based approaches, our algorithms do not require reversibility or exact gradients, and they preserve the structure of the underlying Markov chain. In the finite-sum setting, quantum mean estimation combined with classical variance-reduction techniques improves the stochastic gradient-query complexity for the approximate sampling task. In the stochastic zeroth-order setting, we develop gradient estimators robust to noisy function evaluations, yielding improved evaluation complexity for LMC and HMC. These results apply to strongly log-concave and/or non-log-concave distributions satisfying a log-Sobolev inequality, with convergence guarantees in Wasserstein distance and Kullback--Leibler divergence. We also show that faster sampling methods lead to quantum speedups for optimization, including for non-smooth and approximately convex objectives.
Gibbs state preparation has emerged as a potential avenue for quantum advantage in the study of physical systems. Although there is known advantage for computing local properties of worst-case Gibbs states, recent results suggest a different picture for average-case local systems: at temperatures where their Gibbs states can be efficiently prepared quantumly, quasi-polynomial time classical algorithms can also compute local expectation values. Yet this leaves open the possibility of an advantage under a stronger notion of simulation: sampling. Gibbs sampling underlies Boltzmann machine-based quantum learning algorithms and quantum supremacy proposals, but its classical average-case complexity for physically relevant models remains unresolved. Here, we give strong evidence against an exponential quantum advantage in Gibbs sampling for an ensemble of local systems known as the quantum p-spin model. We describe a classical algorithm for sampling from the Gibbs state to any polynomially small error in total variation distance. Our algorithm is based on a meta-algorithm known as algorithmic stochastic localization (ASL). Our main technical result is a generalized convergence theorem for ASL which reduces approximate sampling from any distribution on the hypercube to estimating the mean of a related "tilted distribution" to sufficiently low additive error. We then extend known quasi-polynomial time algorithms for computing local expectation values of a related disordered model to compute these "tilted means" for the quantum p-spin model. Finally, we give a non-rigorous physics argument that this mean-estimation algorithm works down to the spin glass transition temperature. As efficient quantum algorithms also fail at this transition, this suggests they cannot Gibbs sample the quantum p-spin model at temperatures below that for which classical algorithms are also (almost) efficient.
We study the problem of learning the Hamiltonian of a quantum many-body system given samples from its Gibbs (thermal) state. The classical analog of this problem, known as learning graphical models or Boltzmann machines, is a well-studied question in machine learning and statistics. In this work, we give the first sample-efficient algorithm for the quantum Hamiltonian learning problem. In particular, we prove that polynomially many samples in the number of particles (qudits) are necessary and sufficient for learning the parameters of a spatially local Hamiltonian in l_2-norm. Our main contribution is in establishing the strong convexity of the log-partition function of quantum many-body systems, which along with the maximum entropy estimation yields our sample-efficient algorithm. Classically, the strong convexity for partition functions follows from the Markov property of Gibbs distributions. This is, however, known to be violated in its exact form in the quantum case. We introduce several new ideas to obtain an unconditional result that avoids relying on the Markov property of quantum systems, at the cost of a slightly weaker bound. In particular, we prove a lower bound on the variance of quasi-local operators with respect to the Gibbs state, which might be of independent interest. Our work paves the way toward a more rigorous application of machine learning techniques to quantum many-body problems.