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Automatic structural optimization of tree tensor networks

Toshiya Hikihara
(
- Gunma U., Maebashi
)
,
Hiroshi Ueda
(
- Osaka U. and
- JST, PRESTO and
- RIKEN AICS, Kobe
)
,
Kouichi Okunishi
(
- Niigata U.
)
,
Kenji Harada
(
- Kyoto U.
)
,
Tomotoshi Nishino
(
- Kobe U.
)

Sep 7, 2022

12 pages

Published in:

Phys.Rev.Res. 5 (2023) 1, 013031

Published: Jan 23, 2023

e-Print:

2209.03196 [cond-mat.stat-mech]

DOI:

10.1103/PhysRevResearch.5.013031 (publication)

View in:

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Abstract: (APS)

The tree tensor network (TTN) provides an essential theoretical framework for the practical simulation of quantum many-body systems, where the network structure defined by the connectivity of the isometry tensors plays a crucial role in improving its approximation accuracy. In this paper, we propose a TTN algorithm that enables us to automatically optimize the network structure by local reconnections of isometries to suppress the bipartite entanglement entropy on their legs. The algorithm can be seamlessly implemented to such a conventional TTN approach as the density-matrix renormalization group. We apply the algorithm to the inhomogeneous antiferromagnetic Heisenberg spin chain, having a hierarchical spatial distribution of the interactions. We then demonstrate that the entanglement structure embedded in the ground state of the system can be efficiently visualized as a perfect binary tree in the optimized TTN. Possible improvements and applications of the algorithm are also discussed.

Note:

12 pages, 11 figures, 2 tables, v2: accepted version, to appear in Phys. Rev. Research

density matrix: renormalization group
spin: chain
entropy: entanglement
network
tensor
isometry
binary
Heisenberg
hierarchy
ground state

References(77)

Figures(11)

[1]

Developments in the Tensor Network — from Statistical Mechanics to Quantum Entanglement

Kouichi Okunishi
(
- Niigata U.
)
,
Tomotoshi Nishino
(
- Kobe U.
)
,
Hiroshi Ueda
(
- Osaka U. and
- JST, PRESTO and
- RIKEN AICS, Kobe
)

- J.Phys.Soc.Jap. 91 (2022) 6, 062001,
- J.Phys.Soc.Jap. 91 (2022) 062001
•
e-Print:
- 2111.12223
•
DOI:
- 10.7566/JPSJ.91.062001

[2]

Geometric interpretation of the multi-scale entanglement renormalization ansatz

e-Print:
- 1812.00529

[3]

Tensor networks for complex quantum systems

Román Orús
(
- Donostia Intl. Phys. Ctr., San Sebastian and
- IKERBASQUE, Bilbao
)

- APS Physics 1 (2019) 538-550
•
e-Print:
- 1812.04011
•
DOI:
- 10.1038/s42254-019-0086-7

[4]

Supervised learning with tensor networks, in Advances in Neural Information Processing Systems 29: Annual Conference on Neural Information Processing Systems, December 5–10, 2016, Barcelona, Spain, edited by , pp. 4799–4807, https://researchr.org/publication/StoudenmireS16

D.D. Lee
,
M. Sugiyama
,
U.V. Luxburg
,
I. Guyon
,
R. Garnett

et al.

[5]

Matrix product states and projected entangled pair states: Concepts, symmetries, theorems

J. Ignacio Cirac
(
- Munich, Max Planck Inst. Quantenopt. and
- MCQST, Munich
)
,
David Perez-Garcia
(
- Madrid U. and
- Madrid, ICMAT
)
,
Norbert Schuch
(
)
,
Frank Verstraete
(
- Gent U.
)

- Rev.Mod.Phys. 93 (2021) 4, 045003
•
e-Print:
- 2011.12127
•
DOI:
- 10.1103/RevModPhys.93.045003

[6]

Dimers on a Rectangular Lattice

R.J. Baxter

- J.Math.Phys. 9 (1968) 4, 650
•
DOI:
- 10.1063/1.1664623

[7]

Density matrix formulation for quantum renormalization groups

Steven R. White
(
- UC, Irvine
)

- Phys.Rev.Lett. 69 (1992) 2863-2866
•
DOI:
- 10.1103/PhysRevLett.69.2863

[8]

Density-matrix algorithms for quantum renormalization groups

Steven R. White
(
- UC, Irvine
)

- Phys.Rev.B 48 (1993) 10345-10356
•
DOI:
- 10.1103/PhysRevB.48.10345

[9]

Thermodynamic Limit of Density Matrix Renormalization for the spin-1 heisenberg chain

Stellan Ostlund
(
- Chalmers U. Tech.
)
,
Stefan Rommer
(
- Chalmers U. Tech.
)

- Phys.Rev.Lett. 75 (1995) 3537-3540
•
e-Print:
- cond-mat/9503107
•
DOI:
- 10.1103/PhysRevLett.75.3537

[10]

Class of ansatz wave functions for one-dimensional spin systems and their relation to the density matrix renormalization group

Stefan Rommer
(
- Chalmers U. Tech.
)
,
Stellan Ostlund
(
- Chalmers U. Tech.
)

- Phys.Rev.B 55 (1997) 2164-2181
•
e-Print:
- cond-mat/9606213
•
DOI:
- 10.1103/PhysRevB.55.2164

[11]

Two-dimensional tensor product variational formulation

T. Nishino
,
Y. Hieida
,
K. Okunishi
,
N. Maeshima
,
Y. Akutsu

et al.

- Prog.Theor.Phys. 105 (2001) 409
•
DOI:
- 10.1143/PTP.105.409

[12]

Stable optimization of a tensor product variational state

A. Gendiar
,
N. Maeshima
,
T. Nishino

- Prog.Theor.Phys. 110 (2003) 691
•
DOI:
- 10.1143/PTP.110.691

[13]

Valence-bond states for quantum computation

- Phys.Rev.A 70 (2004) 6, 060302
•
DOI:
- 10.1103/PhysRevA.70.060302

[14]

Criticality, the area law, and the computational power of PEPS

- Phys.Rev.Lett. 96 (2006) 220601,
- Phys.Rev.Lett. 96 (2006) 22, 220601
•
e-Print:
- quant-ph/0601075
•
DOI:
- 10.1103/PhysRevLett.96.220601

[15]

Entanglement Renormalization

G. Vidal
(
- Queensland U. and
- Caltech, IQI
)

- Phys.Rev.Lett. 99 (2007) 22, 220405
•
e-Print:
- cond-mat/0512165
•
DOI:
- 10.1103/PhysRevLett.99.220405

[16]

Algorithms for entanglement renormalization

- Phys.Rev.B 79 (2009) 144108,
- Phys.Rev.B 79 (2009) 14, 144108
•
e-Print:
- 0707.1454
•
DOI:
- 10.1103/PhysRevB.79.144108

[17]

Efficient tree tensor network states (TTNS) for quantum chemistry: Generalizations of the density matrix renormalization group algorithm

N. Nakatani
,
G. Kin-Lic Chan

- J.Chem.Phys. 138 (2013) 134113
•
DOI:
- 10.1063/1.4798639

[18]

Simulating strongly correlated quantum systems with tree tensor networks

- Phys.Rev.B 82 (2010) 20, 205105
•
DOI:
- 10.1103/PhysRevB.82.205105

[19]

Tree tensor network state with variable tensor order: An efficient multireference method for strongly correlated systems

V. Murg
,
F. Verstraete
,
R. Schneider
,
P.R. Nagy
,
Ö. Legeza

- J. Chem. Theory Comput. 11 (2015) 1027
•
DOI:
- 10.1021/ct501187j

[20]

T3NS: Three-legged tree tensor network states

K. Gunst
,
F. Verstraete
,
S. Wouters
,
Ö. Legeza
,
D. Van Neck

- J. Chem. Theory Comput. 14 (2018) 2026
•
DOI:
- 10.1021/acs.jctc.8b00098

[21]

Three-legged tree tensor networks with

K. Gunst
,
F. Verstraete
,
D. Van Neck

- J. Chem. Theory Comput. 15 (2019) 2996
•
DOI:
- 10.1021/acs.jctc.9b00071

[22]

Computing vibrational eigenstates with tree tensor network states (TTNS)

H.R. Larsson

- J.Chem.Phys. 151 (2019) 204102
•
DOI:
- 10.1063/1.5130390

[23]

Quantum transverse-field Ising model on an infinite tree from matrix product states

D. Nagaj
,
E. Farhi
,
J. Goldstone
,
P. Shor
,
I. Sylvester

- Phys.Rev.B 77 (2008) 214431
•
DOI:
- 10.1103/PhysRevB.77.214431

[24]

Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law

- Phys.Rev.B 80 (2009) 235127
•
e-Print:
- 0903.5017
•
DOI:
- 10.1103/PhysRevB.80.235127

[25]

Homogeneous binary trees as ground states of quantum critical Hamiltonians

P. Silvi
,
V. Giovannetti
,
S. Montangero
,
M. Rizzi
,
J.I. Cirac

et al.

- Phys.Rev.A 81 (2010) 062335
•
DOI:
- 10.1103/PhysRevA.81.062335

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