  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.
