An Efficient Algorithm for approximating 1D Ground States

TL;DR

This paper introduces a provably efficient classical algorithm for approximating ground states of 1D quantum systems, providing theoretical guarantees where previous heuristic methods like DMRG lacked proofs.

Contribution

The paper presents the first efficient algorithm with provable guarantees for approximating 1D ground states, improving upon heuristic methods like DMRG.

Findings

Algorithm finds a Matrix Product State with energy close to the ground state

Running time is exponential in bond dimension D, efficient for small D

Exact algorithm for 1D local commuting Hamiltonians

Abstract

The DMRG method is very effective at finding ground states of 1D quantum systems in practice, but it is a heuristic method, and there is no known proof for when it works. In this paper we describe an efficient classical algorithm which provably finds a good approximation of the ground state of 1D systems under well defined conditions. More precisely, our algorithm finds a Matrix Product State of bond dimension $D$ whose energy approximates the minimal energy such states can achieve. The running time is exponential in D, and so the algorithm can be considered tractable even for D which is logarithmic in the size of the chain. The result also implies trivially that the ground state of any local commuting Hamiltonian in 1D can be approximated efficiently; we improve this to an exact algorithm.