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Published July 16, 2010 | Published
Journal Article Open

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.

Additional Information

© 2010 The American Physical Society. Received 17 November 2009; published 16 July 2010. We thank B. Horstmann, A. Kay, D. P´erez-Garc´ıa, and K. G. Vollbrecht for helpful discussions and comments. This work has been supported by the EU (QUEVADIS, SCALA), the German cluster of excellence project MAP, the Gordon and Betty Moore Foundation through Caltech's Center for the Physics of Information, and the National Science Foundation under Grant No. PHY-0803371.

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