Parallel Sampling from the Ising $p$-Spin Model

2026-07-14

AI Summary

This work addresses the problem of efficient parallel sampling for the mean-field p-spin Ising spin glass model on the hypercube at high temperatures. Two novel algorithms are proposed: the first combines block dynamics with an approximate rejection sampler based on an Ising proposal distribution, producing samples within total variation distance ε in parallel time $n^{1/3}\text{polylog}(n/\varepsilon)$; the second parallelizes the annealed stochastic localization (ASL) process via Picard iteration, achieving normalized 2-Wasserstein distance ε in $\text{polylog}(n/\varepsilon)$ time. These methods constitute the first efficient parallel samplers for this model, offering a doubly exponential speedup in runtime and an exponential improvement in total work compared to the naive ASL approach.

Abstract

We study the parallel complexity of sampling from the high-temperature Ising mixed $p$-spin Gibbs measure, a canonical instance of a mean-field spin glass on the hypercube $\{\pm 1\}^n$. We propose two different algorithms for this problem, corresponding to two different regimes of accuracy. Our first algorithm is a parallel implementation of a Markov chain known as block dynamics, combined with an approximate rejection sampling step that uses an Ising model in a novel way as a proposal distribution to approximate the quadratic interaction terms of the $p$-spin Hamiltonian. For any $\varepsilon > 0$, this algorithm runs in $n^{\tfrac{1}{3}}\operatorname{polylog}(\tfrac{n}{\varepsilon})$ parallel time with $\operatorname{poly}(n, \log(\tfrac{1}{\varepsilon}))$ work, and outputs a sample whose law is $\varepsilon$-close to the $p$-spin measure in total variation distance. Our second algorithm uses Picard iterations to parallelize the Algorithmic Stochastic Localization (ASL) process of El Alaoui, Montanari, and Sellke (2025), and for any $\varepsilon > \varepsilon_n$, takes $\operatorname{polylog}(\tfrac{n}{\varepsilon})$ parallel time and $\operatorname{poly}(\tfrac{n}{\varepsilon})$ work to produce a sample that is $\varepsilon$-close to the $p$-spin measure in the normalized 2-Wasserstein metric. Here, $\varepsilon_n > 0$ is a threshold that goes to $0$ as $n \to \infty$. Our result constitutes a doubly exponential improvement in the $\varepsilon$ dependence of the runtime and an exponential improvement in the $\varepsilon$ dependence of the total work when compared to naïve ASL, whose runtime scales as $\exp(\operatorname{poly}(\tfrac{1}{\varepsilon}))$.

Problem

• Ising p-spin model
• parallel sampling
• spin glass
• Gibbs measure
• parallel complexity

Innovation

• parallel sampling
• p-spin model
• block dynamics
• Algorithmic Stochastic Localization
• Picard iteration