A polynomial-time algorithm for the ground state of 1D gapped local Hamiltonians

TL;DR

This paper presents the first randomized polynomial-time algorithm to approximate ground states of 1D gapped local Hamiltonians using matrix product states, making the computation of such states provably efficient.

Contribution

It introduces a novel polynomial-time algorithm for 1D gapped Hamiltonians that produces accurate matrix product state approximations, leveraging structural insights and convex programming.

Findings

Algorithm runs in polynomial time

Produces inverse-polynomial accuracy approximations

Applicable to a major class of Hamiltonians

Abstract

Computing ground states of local Hamiltonians is a fundamental problem in condensed matter physics. We give the first randomized polynomial-time algorithm for finding ground states of gapped one-dimensional Hamiltonians: it outputs an (inverse-polynomial) approximation, expressed as a matrix product state (MPS) of polynomial bond dimension. The algorithm combines many ingredients, including recently discovered structural features of gapped 1D systems, convex programming, insights from classical algorithms for 1D satisfiability, and new techniques for manipulating and bounding the complexity of MPS. Our result provides one of the first major classes of Hamiltonians for which computing ground states is provably tractable despite the exponential nature of the objects involved.