Isometric Tensor Network States in Two Dimensions
Abstract
Tensor-network states (TNS) are a promising but numerically challenging tool for simulating two-dimensional (2D) quantum many-body problems. We introduce an isometric restriction of the TNS ansatz that allows for highly efficient contraction of the network. Here we consider two concrete applications using this ansatz. First, we show that a matrix-product state representation of a 2D quantum state can be iteratively transformed into an isometric 2D TNS. Second, we introduce a 2D version of the time-evolving block decimation algorithm for approximating of the ground state of a Hamiltonian as an isometric TNS-which we demonstrate for the 2D transverse field Ising model.
- Authors:
-
- Univ. of California, Berkeley, CA (United States)
- Technische Univ. Munchen, Garching (Germany); Munich Center for Quantum Science and Technology (MCQST), Munchen (Germany)
- Publication Date:
- Research Org.:
- Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Materials Sciences & Engineering Division; German Research Foundation (DFG); European Union (EU); National Science Foundation (NSF)
- OSTI Identifier:
- 1603571
- Grant/Contract Number:
- AC02-05CH11231; PO 1370/2-1; TRR80; 771537; PHY-1607611
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Physical Review Letters
- Additional Journal Information:
- Journal Volume: 124; Journal Issue: 3; Journal ID: ISSN 0031-9007
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY
Citation Formats
Zaletel, Michael P., and Pollmann, Frank. Isometric Tensor Network States in Two Dimensions. United States: N. p., 2020.
Web. doi:10.1103/physrevlett.124.037201.
Zaletel, Michael P., & Pollmann, Frank. Isometric Tensor Network States in Two Dimensions. United States. https://doi.org/10.1103/physrevlett.124.037201
Zaletel, Michael P., and Pollmann, Frank. Wed .
"Isometric Tensor Network States in Two Dimensions". United States. https://doi.org/10.1103/physrevlett.124.037201. https://www.osti.gov/servlets/purl/1603571.
@article{osti_1603571,
title = {Isometric Tensor Network States in Two Dimensions},
author = {Zaletel, Michael P. and Pollmann, Frank},
abstractNote = {Tensor-network states (TNS) are a promising but numerically challenging tool for simulating two-dimensional (2D) quantum many-body problems. We introduce an isometric restriction of the TNS ansatz that allows for highly efficient contraction of the network. Here we consider two concrete applications using this ansatz. First, we show that a matrix-product state representation of a 2D quantum state can be iteratively transformed into an isometric 2D TNS. Second, we introduce a 2D version of the time-evolving block decimation algorithm for approximating of the ground state of a Hamiltonian as an isometric TNS-which we demonstrate for the 2D transverse field Ising model.},
doi = {10.1103/physrevlett.124.037201},
journal = {Physical Review Letters},
number = 3,
volume = 124,
place = {United States},
year = {Wed Jan 01 00:00:00 EST 2020},
month = {Wed Jan 01 00:00:00 EST 2020}
}
Web of Science
Works referenced in this record:
The density-matrix renormalization group in the age of matrix product states
journal, January 2011
- Schollwöck, Ulrich
- Annals of Physics, Vol. 326, Issue 1
Corner Transfer Matrix Renormalization Group Method
journal, April 1996
- Nishino, Tomotoshi; Okunishi, Kouichi
- Journal of the Physical Society of Japan, Vol. 65, Issue 4
Matrix product state renormalization
journal, November 2016
- Bal, M.; Rams, M. M.; Zauner, V.
- Physical Review B, Vol. 94, Issue 20
An area law for one-dimensional quantum systems
journal, August 2007
- Hastings, M. B.
- Journal of Statistical Mechanics: Theory and Experiment, Vol. 2007, Issue 08
Tensor network study of the Shastry-Sutherland model in zero magnetic field
journal, March 2013
- Corboz, Philippe; Mila, Frédéric
- Physical Review B, Vol. 87, Issue 11
The entanglement of purification
journal, September 2002
- Terhal, Barbara M.; Horodecki, Michał; Leung, Debbie W.
- Journal of Mathematical Physics, Vol. 43, Issue 9
Finding purifications with minimal entanglement
journal, December 2018
- Hauschild, Johannes; Leviatan, Eyal; Bardarson, Jens H.
- Physical Review B, Vol. 98, Issue 23
Efficient Simulation of One-Dimensional Quantum Many-Body Systems
journal, July 2004
- Vidal, Guifré
- Physical Review Letters, Vol. 93, Issue 4
String Order and Symmetries in Quantum Spin Lattices
journal, April 2008
- Pérez-García, D.; Wolf, M. M.; Sanz, M.
- Physical Review Letters, Vol. 100, Issue 16
Continuous Matrix Product States for Quantum Fields
journal, May 2010
- Verstraete, F.; Cirac, J. I.
- Physical Review Letters, Vol. 104, Issue 19
Real-space parallel density matrix renormalization group
journal, April 2013
- Stoudenmire, E. M.; White, Steven R.
- Physical Review B, Vol. 87, Issue 15
Accurate Determination of Tensor Network State of Quantum Lattice Models in Two Dimensions
journal, August 2008
- Jiang, H. C.; Weng, Z. Y.; Xiang, T.
- Physical Review Letters, Vol. 101, Issue 9
Corner Transfer Matrix Algorithm for Classical Renormalization Group
journal, October 1997
- Nishino, Tomotoshi; Okunishi, Kouichi
- Journal of the Physical Society of Japan, Vol. 66, Issue 10
Sequentially generated states for the study of two-dimensional systems
journal, May 2008
- Bañuls, M. C.; Pérez-García, D.; Wolf, M. M.
- Physical Review A, Vol. 77, Issue 5
Matrix product states represent ground states faithfully
journal, March 2006
- Verstraete, F.; Cirac, J. I.
- Physical Review B, Vol. 73, Issue 9
Coarse-graining renormalization by higher-order singular value decomposition
journal, July 2012
- Xie, Z. Y.; Chen, J.; Qin, M. P.
- Physical Review B, Vol. 86, Issue 4
Variational optimization with infinite projected entangled-pair states
journal, July 2016
- Corboz, Philippe
- Physical Review B, Vol. 94, Issue 3
Density matrix formulation for quantum renormalization groups
journal, November 1992
- White, Steven R.
- Physical Review Letters, Vol. 69, Issue 19
Algorithms for Entanglement Renormalization: Boundaries, Impurities and Interfaces
journal, April 2014
- Evenbly, G.; Vidal, G.
- Journal of Statistical Physics, Vol. 157, Issue 4-5
Vertical density matrix algorithm: A higher-dimensional numerical renormalization scheme based on the tensor product state ansatz
journal, June 2001
- Maeshima, Nobuya; Hieida, Yasuhiro; Akutsu, Yasuhiro
- Physical Review E, Vol. 64, Issue 1
Gapless Spin-Liquid Ground State in the Kagome Antiferromagnet
journal, March 2017
- Liao, H. J.; Xie, Z. Y.; Chen, J.
- Physical Review Letters, Vol. 118, Issue 13
Unifying time evolution and optimization with matrix product states
journal, October 2016
- Haegeman, Jutho; Lubich, Christian; Oseledets, Ivan
- Physical Review B, Vol. 94, Issue 16
Equivalence of the variational matrix product method and the density matrix renormalization group applied to spin chains
journal, August 1998
- Dukelsky, J.; Martín-Delgado, M. A.; Nishino, T.
- Europhysics Letters (EPL), Vol. 43, Issue 4
Thermodynamic Limit of Density Matrix Renormalization
journal, November 1995
- Östlund, Stellan; Rommer, Stefan
- Physical Review Letters, Vol. 75, Issue 19
Faster methods for contracting infinite two-dimensional tensor networks
journal, December 2018
- Fishman, M. T.; Vanderstraeten, L.; Zauner-Stauber, V.
- Physical Review B, Vol. 98, Issue 23
Gradient methods for variational optimization of projected entangled-pair states
journal, October 2016
- Vanderstraeten, Laurens; Haegeman, Jutho; Corboz, Philippe
- Physical Review B, Vol. 94, Issue 15
Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems
journal, March 2008
- Verstraete, F.; Murg, V.; Cirac, J. I.
- Advances in Physics, Vol. 57, Issue 2
Tensor Renormalization Group Approach to Two-Dimensional Classical Lattice Models
journal, September 2007
- Levin, Michael; Nave, Cody P.
- Physical Review Letters, Vol. 99, Issue 12
Stripe order in the underdoped region of the two-dimensional Hubbard model
journal, November 2017
- Zheng, Bo-Xiao; Chung, Chia-Min; Corboz, Philippe
- Science, Vol. 358, Issue 6367
Second Renormalization of Tensor-Network States
journal, October 2009
- Xie, Z. Y.; Jiang, H. C.; Chen, Q. N.
- Physical Review Letters, Vol. 103, Issue 16
Classical Simulation of Infinite-Size Quantum Lattice Systems in Two Spatial Dimensions
journal, December 2008
- Jordan, J.; Orús, R.; Vidal, G.
- Physical Review Letters, Vol. 101, Issue 25
Works referencing / citing this record:
Simulating groundstate and dynamical quantum phase transitions on a superconducting quantum computer
journal, October 2022
- Dborin, James; Wimalaweera, Vinul; Barratt, F.
- Nature Communications, Vol. 13, Issue 1
Three-dimensional isometric tensor networks
journal, June 2021
- Tepaske, Maurits S. J.; Luitz, David J.
- Physical Review Research, Vol. 3, Issue 2
Holographic quantum algorithms for simulating correlated spin systems
journal, July 2021
- Foss-Feig, Michael; Hayes, David; Dreiling, Joan M.
- Physical Review Research, Vol. 3, Issue 3
Riemannian optimization of isometric tensor networks
journal, January 2021
- Hauru, Markus; Van Damme, Maarten; Haegeman, Jutho
- SciPost Physics, Vol. 10, Issue 2
Isometric Tensor Network representation of string-net liquids
text, January 2019
- Soejima, Tomohiro; Siva, Karthik; Bultinck, Nick
- arXiv
On Stability of Tensor Networks and Canonical Forms
preprint, January 2020
- Zhang, Yifan; Solomonik, Edgar
- arXiv