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

• URL: http://arxiv.org/abs/2307.11053v1
• Date: Thu, 20 Jul 2023 17:34:06 GMT
• Title: Random insights into the complexity of two-dimensional tensor network calculations
• Authors: Sofia Gonzalez-Garcia, Shengqi Sang, Timothy H. Hsieh, Sergio Boixo, Guifre Vidal, Andrew C. Potter and Romain Vasseur
• Abstract summary: Projected entangled pair states offer memory-efficient representations of some quantum many-body states. Projected entangled pair states offer memory-efficient representations of some quantum many-body states that obey an entanglement area law.
• Score: 0.27708222692419743
• License: http://creativecommons.org/licenses/by/4.0/
• 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

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