Tensor networks for $p$-spin models

TL;DR

This paper presents a tensor network algorithm with lossless bond compression for solving $p$-spin models, efficiently handling different phases and improving contraction scaling by emulating classical algorithms and reducing redundancies.

Contribution

The authors introduce a novel tensor network method that employs lossless bond compression to efficiently solve $p$-spin models across different regimes, unifying classical and quantum approaches.

Findings

Bond compression emulates leaf-removal algorithm.

Efficiently solves the easy phase of $p$-spin models.

Achieves subexponential contraction scaling in core instances.

Abstract

We introduce a tensor network algorithm for the solution of $p$-spin models. We show that bond compression through rank-revealing decompositions performed during the tensor network contraction resolves logical redundancies in the system exactly and is thus lossless, yet leads to qualitative changes in runtime scaling in different regimes of the model. First, we find that bond compression emulates the so-called leaf-removal algorithm, solving the problem efficiently in the "easy" phase. Past a dynamical phase transition, we observe superpolynomial runtimes, reflecting the appearance of a core component. We then develop a graphical method to study the scaling of contraction for a minimal ensemble of core-only instances. We find subexponential scaling, improving on the exponential scaling that occurs without compression. Our results suggest that our tensor network algorithm subsumes the classical leaf removal algorithm and simplifies redundancies in the $p$-spin model through lossless compression, all without explicit knowledge of the problem's structure.