Projected Entangled Pair States: Fundamental analytical and numerical limitations

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.

arXiv commentv1: 4+3 pages, 5 figures. v2: extensively rewritten

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Cited by in corpus (16)

• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Augmenting Density Matrix Renormalization Group with Disentanglers
• Optimization at the boundary of the tensor network variety
• On the closedness and geometry of tensor network state sets
• The minimal canonical form of a tensor network
• Charting the space of ground states with tensor networks
• Undecidability in Physics: a Review
• Many bounded versions of undecidable problems are NP-hard
• Dual-isometric Projected Entangled Pair States
• The resource theory of tensor networks
• A generic quantum Wielandt's inequality
• Two Dimensional Isometric Tensor Networks on an Infinite Strip
• Internal structure of gauge-invariant Projected Entangled Pair States
• Sequential Generation of Two-dimensional Super-area-law States with Local Parent Hamiltonian
• Equivalence of residual entropy of hexagonal and cubic ices from tensor network methods
• Undecidability of the spectral gap in rotationally symmetric Hamiltonians