An Efficient Algorithm for approximating 1D Ground States

Details An Efficient Algorithm for approximating 1D Ground States An Efficient Algorithm for approximating 1D Ground States

2009-10-27

arxiv.org/abs/0910.5055v4

Phys. Rev. A 82, 012315 (2010)

Physics

Quantum Physics

16 pages, 11 figures. Replaced with updated version + link to PRA journal

Scientific paper

10.1103/PhysRevA.82.012315

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.

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