Projected Entangled Pair States: Fundamental analytical and numerical limitations

TL;DR

This paper demonstrates that key analytical and numerical problems in Projected Entangled Pair States (PEPS) are algorithmically undecidable, revealing fundamental limitations in understanding quantum many-body systems with PEPS.

Contribution

It proves that essential questions about symmetries and stabilization in PEPS are undecidable, highlighting intrinsic theoretical constraints.

Findings

Key problems in PEPS are algorithmically undecidable.

Fundamental limitations exist in fully understanding PEPS.

Highlights the difference between MPS and PEPS in terms of solvability.

Abstract

Matrix Product States (MPS) and Projected Entangled Pair States (PEPS) are powerful analytical and numerical tools to assess quantum many-body systems in one and higher dimensions, respectively. While MPS are comprehensively understood, in PEPS fundamental questions, relevant analytically as well as numerically, remain open, such as how to encode symmetries in full generality, or how to stabilize numerical methods using canonical forms. Here, we show that these key problems, as well as a number of related questions, are algorithmically undecidable, that is, they cannot be fully resolved in a systematic way. Our work thereby exposes fundamental limitations to a full and unbiased understanding of quantum many-body systems using PEPS.