Approximating Gibbs states of local Hamiltonians efficiently with projected entangled pair states

Andras Molnar, Norbert Schuch, Frank Verstraete, and J. Ignacio Cirac
Phys. Rev. B 91, 045138 – Published 29 January 2015

Abstract

We analyze the error of approximating Gibbs states of local quantum spin Hamiltonians on lattices with projected entangled pair states (PEPS) as a function of the bond dimension (D), temperature (β1), and system size (N). First, we introduce a compression method in which the bond dimension scales as D=eO(log22(N/ε)) if β<O(log2N). Second, building on the work of Hastings [M. B. Hastings, Phys. Rev. B 73, 085115 (2006)], we derive a polynomial scaling relation, D=(N/ε)O(β). This implies that the manifold of PEPS forms an efficient representation of Gibbs states of local quantum Hamiltonians. From those bounds it also follows that ground states can be approximated with D=NO(log2N) whenever the density of states only grows polynomially in the system size. All results hold for any spatial dimension of the lattice.

  • Figure
  • Received 29 September 2014
  • Revised 23 December 2014

DOI:https://doi.org/10.1103/PhysRevB.91.045138

©2015 American Physical Society

Authors & Affiliations

Andras Molnar1, Norbert Schuch2, Frank Verstraete3, and J. Ignacio Cirac1

  • 1Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany
  • 2JARA Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany
  • 3Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria and Department of Physics and Astronomy, Ghent University, Ghent, Belgium

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Vol. 91, Iss. 4 — 15 January 2015

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