TY - JOUR

T1 - Efficient algorithm for approximating one-dimensional ground states

AU - Aharonov, Dorit

AU - Arad, Itai

AU - Irani, Sandy

PY - 2010/7/16

Y1 - 2010/7/16

N2 - The density-matrix renormalization-group method is very effective at finding ground states of one-dimensional (1D) quantum systems in practice, but it is a heuristic method, and there is no known proof for when it works. In this article 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.

AB - The density-matrix renormalization-group method is very effective at finding ground states of one-dimensional (1D) quantum systems in practice, but it is a heuristic method, and there is no known proof for when it works. In this article 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.

UR - http://www.scopus.com/inward/record.url?scp=77954849666&partnerID=8YFLogxK

U2 - 10.1103/PhysRevA.82.012315

DO - 10.1103/PhysRevA.82.012315

M3 - Article

AN - SCOPUS:77954849666

SN - 1050-2947

VL - 82

JO - Physical Review A - Atomic, Molecular, and Optical Physics

JF - Physical Review A - Atomic, Molecular, and Optical Physics

IS - 1

M1 - 012315

ER -