Approximating local observables on projected entangled pair states

TL;DR

This paper demonstrates that for certain projected entangled pair states, local expectation values can be computed efficiently, providing theoretical justification for the success of numerical methods and exploring quantum computing applications.

Contribution

It shows that local expectation values for PEPS approximating ground states can be computed in quasi-polynomial time, bridging theory and practical algorithms.

Findings

Local expectation values are computable in quasi-polynomial time for PEPS.

Transfer operators of PEPS have a spectral gap, indicating stability.

Quantum algorithms can be used to compute local observables on small-scale quantum computers.

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.