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Parallel Sampling from the Ising p-Spin Model

Authors Nima Anari , Aniket Das , Alireza Haqi

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ACM Subject Classification
  • Theory of computation → Parallel algorithms
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Probabilistic algorithms
Keywords
  • spin glasses
  • parallel sampling
  • Glauber dynamics
  • stochastic localization

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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 {±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 ε > 0, this algorithm runs in n^{1/3} polylog(n/ε) parallel time with poly(n/ε) work, and outputs a sample whose law is ε-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 ε > ε_n, takes polylog(n/ε) parallel time and poly(n/ε) work to produce a sample that is ε-close to the p-spin measure in the normalized 2-Wasserstein metric. Here, ε_n > 0 is a threshold that goes to 0 as n → ∞. Our result constitutes a doubly exponential improvement in the ε dependence of the runtime and an exponential improvement in the ε dependence of the total work when compared to naïve ASL, whose runtime scales as exp(poly(1/ε)).

Cite As Get BibTex

Nima Anari, Aniket Das, and Alireza Haqi. Parallel Sampling from the Ising p-Spin Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 71:1-71:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.71

Author Details

Nima Anari
  • Stanford University, CA, USA
Aniket Das
  • Stanford University, CA, USA
Alireza Haqi
  • Stanford University, CA, USA

Funding

This work is supported by funds from NSF-2125511, NSF CCF-2045354, NSF MAI-2501597.

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