An Efficient Algorithm for approximating 1D Ground States

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.