Efficient algorithm for approximating one-dimensional ground states

Journal Article · Thu Jul 15 00:00:00 EDT 2010 · Physical Review. A

DOI:https://doi.org/10.1103/PHYSREVA.82.012315· OSTI ID:21442888

Aharonov, Dorit; Arad, Itai; Irani, Sandy ^[1]

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1. School of Computer Science and Engineering, Hebrew University, Jerusalem (Israel)

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.

• OSTI ID: 21442888
• Journal Information: Physical Review. A, Vol. 82, Issue 1; Other Information: DOI: 10.1103/PhysRevA.82.012315; (c) 2010 The American Physical Society; ISSN 1050-2947
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
74 ATOMIC AND MOLECULAR PHYSICS
ALGORITHMS
APPROXIMATIONS
COMMUTATION RELATIONS
DENSITY MATRIX
GROUND STATES
HAMILTONIANS
ONE-DIMENSIONAL CALCULATIONS
QUANTUM STATES
RENORMALIZATION
CALCULATION METHODS
ENERGY LEVELS
MATHEMATICAL LOGIC
MATHEMATICAL OPERATORS
MATRICES
QUANTUM OPERATORS