Even in low dimensions, sampling from multi-modal distributions is challenging. We provide the first sampling algorithm for a broad class of distributions -- including all Gaussian mixtures -- with a query complexity that is polynomial in the parameters governing multi-modality, assuming fixed dimension. Our sampling algorithm simulates a time-reversed diffusion process, using a self-normalized Monte Carlo estimator of the intermediate score functions. Unlike previous works, it avoids metastability, requires no prior knowledge of the mode locations, and relaxes the well-known log-smoothness assumption which excluded general Gaussian mixtures so far.
Even in low dimensions, sampling from multi-modal distributions is challenging. We provide the first sampling algorithm for a broad class of distributions -- including all Gaussian mixtures -- with a query complexity that is polynomial in the parameters governing multi-modality, assuming fixed dimension. Our sampling algorithm simulates a time-reversed diffusion process, using a self-normalized Monte Carlo estimator of the intermediate score functions. Unlike previous works, it avoids metastability, requires no prior knowledge of the mode locations, and relaxes the well-known log-smoothness assumption which excluded general Gaussian mixtures so far.
We establish the first tight lower bound of Ω(loglogκ) on the query complexity of sampling from the class of strongly log-concave and log-smooth distributions with condition number κ in one dimension. Whereas existing guarantees for MCMC-based algorithms scale polynomially in κ, we introduce a novel algorithm based on rejection sampling that closes this doubly exponential gap.
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.
Abstract We study numerical integration over bounded regions in Rs R s , s≥1 s ≥ 1 , with respect to some probability measure. We replace random sampling with quasi-Monte Carlo methods, where the underlying point set is derived from deterministic constructions which aim to fill the space more evenly than random points. Ordinarily, such quasi-Monte Carlo point sets are designed for the uniform measure, and the theory only works for product measures when a coordinate-wise transformation is applied. Going beyond this setting, we first consider the case where the target density is a mixture distribution where each term in the mixture comes from a product distribution. Next, we consider target densities which can be approximated with such mixture distributions. In order to be able to use an approximation of the target density, we require the approximation to be a sum of coordinate-wise products and that the approximation is positive everywhere (so that they can be re-scaled to probability density functions). We use tensor product hat function approximations for this purpose here, since a hat function approximation of a positive function is itself positive. We also study more complex algorithms, where we first approximate the target density with a general Gaussian mixture distribution and approximate this mixture distribution with an adaptive hat function approximation on rotated intervals. The Gaussian mixture approximation allows us (at least to some degree) to locate the essential parts of the target density, whereas the adaptive hat function approximation allows us to approximate the finer structure of the target density. We prove convergence rates for each of the integration techniques based on quasi-Monte Carlo sampling for integrands with bounded partial mixed derivatives. The employed algorithms are based on digital (t, s)-sequences over the finite field F2 F 2 and an inversion method. Numerical examples illustrate the performance of the algorithms for some target densities and integrands.
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.
Hamiltonian Monte Carlo (HMC) is a Markov chain algorithm for sampling from a high-dimensional distribution with density e−f(x), given access to the gradient of f. A particular case of interest is that of a d-dimensional Gaussian distribution with covariance matrix Σ, in which case f(x)=x⊤Σ−1x. We show that HMC can sample from a distribution that is ε-close in total variation distance using O(κd1/4log(1/ε)) gradient queries, where κ is the condition number of Σ. Our algorithm uses long and random integration times for the Hamiltonian dynamics. This contrasts with (and was motivated by) recent results that give an Ω(κd1/2) query lower bound for HMC with fixed integration times, even for the Gaussian case.
Conventional score-based diffusion models learn scores without representing normalized densities, whereas tractable normalized models support both sampling and direct likelihood evaluation. A recent tensor-network approach provides such a representation but is largely restricted to low-dimensional lattices. Here we introduce a discrete diffusion model that parameterizes normalized probability distributions using variational autoregressive networks. Explicit Markov jump operators govern the forward noising and reverse denoising dynamics, extending discrete diffusion models with normalized distributions to spin systems on higher-dimensional lattices. We apply this framework to the two- and three-dimensional Ising models across ordered, critical, and disordered regimes, accurately computing thermodynamic quantities including free energy, energy, and magnetization. We further integrate the framework with Monte Carlo sampling, using adaptive diffusion steps to maintain high acceptance rates even at low temperatures while enhancing sample diversity. These results establish a neural-network framework for the discrete diffusion model with normalized probability distributions.