Random insights into the complexity of two-dimensional tensor network calculations

TL;DR

This paper analyzes the computational complexity of contracting two-dimensional random PEPS using a statistical mechanics approach, revealing a phase transition in sampling amplitudes but efficient estimation of norms and correlations at finite bond dimensions.

Contribution

It introduces a novel analytical framework based on random matrix theory to study the complexity of 2D PEPS contractions, highlighting a phase transition in sampling and efficiency in estimating physical quantities.

Findings

Sampling wave-function amplitudes undergo a complexity phase transition at a critical bond dimension.

Norms and correlation functions can be efficiently estimated at finite bond dimensions.

Numerical evidence supports the theoretical analysis across various regimes.

Abstract

Projected entangled pair states (PEPS) offer memory-efficient representations of some quantum many-body states that obey an entanglement area law, and are the basis for classical simulations of ground states in two-dimensional (2d) condensed matter systems. However, rigorous results show that exactly computing observables from a 2d PEPS state is generically a computationally hard problem. Yet approximation schemes for computing properties of 2d PEPS are regularly used, and empirically seen to succeed, for a large subclass of (not too entangled) condensed matter ground states. Adopting the philosophy of random matrix theory, in this work we analyze the complexity of approximately contracting a 2d random PEPS by exploiting an analytic mapping to an effective replicated statistical mechanics model that permits a controlled analysis at large bond dimension. Through this statistical-mechanics lens, we argue that: i) although approximately sampling wave-function amplitudes of random PEPS faces a computational-complexity phase transition above a critical bond dimension, ii) one can generically efficiently estimate the norm and correlation functions for any finite bond dimension. These results are supported numerically for various bond-dimension regimes. It is an important open question whether the above results for random PEPS apply more generally also to PEPS representing physically relevant ground states