Random insights into the complexity of two-dimensional tensor network calculations
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
Cite
@article{arxiv.2307.11053,
title = {Random insights into the complexity of two-dimensional tensor network calculations},
author = {Sofia Gonzalez-Garcia and Shengqi Sang and Timothy H. Hsieh and Sergio Boixo and Guifre Vidal and Andrew C. Potter and Romain Vasseur},
journal= {arXiv preprint arXiv:2307.11053},
year = {2024}
}