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Compression of Correlation Matrices and an Efficient Method for Forming Matrix Product States of Fermionic Gaussian States

Fishman, Matthew T. and White, Steven R. (2015) Compression of Correlation Matrices and an Efficient Method for Forming Matrix Product States of Fermionic Gaussian States. Physical Review B, 92 (7). Art. No. 075132. ISSN 1098-0121. doi:10.1103/PhysRevB.92.075132.

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Here we present an efficient and numerically stable procedure for compressing a correlation matrix into a set of local unitary single-particle gates, which leads to a very efficient way of forming the matrix product state (MPS) approximation of a pure fermionic Gaussian state, such as the ground state of a quadratic Hamiltonian. The procedure involves successively diagonalizing subblocks of the correlation matrix to isolate local states which are purely occupied or unoccupied. A small number of nearest-neighbor unitary gates isolate each local state. The MPS of this state is formed by applying the many-body version of these gates to a product state. We treat the simple case of compressing the correlation matrix of spinless free fermions with definite particle number in detail, although the procedure is easily extended to fermions with spin and more general BCS states (utilizing the formalism of Majorana modes). We also present a density matrix renormalization group–like algorithm to obtain the compressed correlation matrix directly from a hopping Hamiltonian. In addition, we discuss a slight variation of the procedure which leads to a simple construction of the multiscale entanglement renormalization ansatz of a fermionic Gaussian state, and present a simple picture of orthogonal wavelet transforms in terms of the gate structure we present in this paper. As a simple demonstration, we analyze the Su-Schrieffer-Heeger model (free fermions on a one-dimensional lattice with staggered hopping amplitudes).

Item Type:Article
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URLURL TypeDescription Paper
White, Steven R.0000-0002-0831-9097
Additional Information:© 2015 American Physical Society. Received 2 June 2015; published 21 August 2015. We would like to thank G. Evenbly for many helpful discussions and comments on the manuscript. We would also like to acknowledge input from M. Zaletel and G. Chan. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1144469. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors(s) and do not necessarily reflect the views of the National Science Foundation. This work was also supported by the Simons Foundation through the many electron collaboration.
Group:Institute for Quantum Information and Matter
Funding AgencyGrant Number
NSF Graduate Research FellowshipDGE-1144469
Simons FoundationUNSPECIFIED
Issue or Number:7
Classification Code:PACS number(s): 71.10.Fd, 03.65.Ud, 02.70.−c
Record Number:CaltechAUTHORS:20150701-092955238
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:58728
Deposited By: Tony Diaz
Deposited On:06 Jul 2015 18:38
Last Modified:10 Nov 2021 22:09

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