Approximating local observables on projected entangled pair states

Abstract

Tensor network states are for good reasons believed to capture ground states of gapped local Hamiltonians arising in the condensed matter context, states which are in turn expected to satisfy an entanglement area law. However, the computational hardness of contracting projected entangled pair states in two and higher dimensional systems is often seen as a significant obstacle when devising higher-dimensional variants of the density-matrix renormalisation group method. In this work, we show that for those projected entangled pair states that are expected to provide good approximations of such ground states of local Hamiltonians, one can compute local expectation values in quasi-polynomial time. We therefore provide a complexity-theoretic justification of why state-of-the-art numerical tools work so well in practice. We comment on how the transfer operators of such projected entangled pair states have a gap and discuss notions of local topological quantum order. We finally turn to the computation of local expectation values on quantum computers, providing a meaningful application for a small-scale quantum computer.

7 pages, 1 figure, minor changes in v2

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• Absence of barren plateaus in finite local-depth circuits with long-range entanglement
• Non-equilibrium phase transitions in $(1+1)$-dimensional quantum cellular automata with controllable quantum correlations
• Entanglement estimation in tensor network states via sampling
• An introduction to infinite projected entangled-pair state methods for variational ground state simulations using automatic differentiation
• Locality at the boundary implies gap in the bulk for 2D PEPS
• Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions -- Part I
• Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions -- Part II
• Circuit-to-Hamiltonian from tensor networks and fault tolerance
• Positive bias makes tensor-network contraction tractable
• Locally accurate tensor networks for thermal states and time evolution