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Title: Matrix product state and mean-field solutions for one-dimensional systems can be found efficiently

Abstract

We consider the problem of approximating ground states of one-dimensional quantum systems within the two most common variational ansatzes, namely, the mean-field ansatz and matrix product states. We show that both for mean field and for matrix product states of fixed bond dimension, the optimal solutions can be found in a way which is provably efficient (i.e., scales polynomially). This implies that the corresponding variational methods can be in principle recast in a way which scales provably polynomially. Moreover, our findings imply that ground states of one-dimensional commuting Hamiltonians can be found efficiently.

Authors:
 [1];  [1]
  1. Max-Planck-Institut fuer Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching (Germany)
Publication Date:
OSTI Identifier:
21440482
Resource Type:
Journal Article
Journal Name:
Physical Review. A
Additional Journal Information:
Journal Volume: 82; Journal Issue: 1; Other Information: DOI: 10.1103/PhysRevA.82.012314; (c) 2010 The American Physical Society; Journal ID: ISSN 1050-2947
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COMMUTATION RELATIONS; GROUND STATES; HAMILTONIANS; MATHEMATICAL SOLUTIONS; MATRICES; MEAN-FIELD THEORY; ONE-DIMENSIONAL CALCULATIONS; QUANTUM STATES; VARIATIONAL METHODS; CALCULATION METHODS; ENERGY LEVELS; MATHEMATICAL OPERATORS; QUANTUM OPERATORS

Citation Formats

Schuch, Norbert, Institute for Quantum Information, California Institute of Technology, MC 305-16, Pasadena, California 91125, and Cirac, J Ignacio. Matrix product state and mean-field solutions for one-dimensional systems can be found efficiently. United States: N. p., 2010. Web. doi:10.1103/PHYSREVA.82.012314.
Schuch, Norbert, Institute for Quantum Information, California Institute of Technology, MC 305-16, Pasadena, California 91125, & Cirac, J Ignacio. Matrix product state and mean-field solutions for one-dimensional systems can be found efficiently. United States. https://doi.org/10.1103/PHYSREVA.82.012314
Schuch, Norbert, Institute for Quantum Information, California Institute of Technology, MC 305-16, Pasadena, California 91125, and Cirac, J Ignacio. 2010. "Matrix product state and mean-field solutions for one-dimensional systems can be found efficiently". United States. https://doi.org/10.1103/PHYSREVA.82.012314.
@article{osti_21440482,
title = {Matrix product state and mean-field solutions for one-dimensional systems can be found efficiently},
author = {Schuch, Norbert and Institute for Quantum Information, California Institute of Technology, MC 305-16, Pasadena, California 91125 and Cirac, J Ignacio},
abstractNote = {We consider the problem of approximating ground states of one-dimensional quantum systems within the two most common variational ansatzes, namely, the mean-field ansatz and matrix product states. We show that both for mean field and for matrix product states of fixed bond dimension, the optimal solutions can be found in a way which is provably efficient (i.e., scales polynomially). This implies that the corresponding variational methods can be in principle recast in a way which scales provably polynomially. Moreover, our findings imply that ground states of one-dimensional commuting Hamiltonians can be found efficiently.},
doi = {10.1103/PHYSREVA.82.012314},
url = {https://www.osti.gov/biblio/21440482}, journal = {Physical Review. A},
issn = {1050-2947},
number = 1,
volume = 82,
place = {United States},
year = {2010},
month = {7}
}