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Tensor Network Methods in Many-body physics

Andras Molnar

Ludwig-Maximilians-Universit¨at Mu¨nchen Max-Planck-Institut fu¨r Quanutenoptik

Mu¨nchen 2019

Tensor Network Methods in Many-body physics

Andras Molnar

Dissertation an der Fakult¨at fu¨r Physik der Ludwig–Maximilians–Universita¨t
Mu¨nchen

vorgelegt von Andras Molnar aus Budapest

Mu¨nchen, den 25/02/2019

Erstgutachter: Prof. Dr. Jan von Delft Zweitgutachter: Prof. Dr. J. Ignacio Cirac Tag der mu¨ndlichen Pru¨fung: 3. Mai 2019

Abstract

Strongly correlated systems exhibit phenomena – such as high-TC superconductivity or the fractional quantum Hall effect – that are not explicable by classical and semi-classical methods. Moreover, due to the exponential scaling of the associated Hilbert space, solving the proposed model Hamiltonians by brute-force numerical methods is bound to fail. Thus, it is important to develop novel numerical and analytical methods that can explain the physics in this regime.
Tensor Network states are quantum many-body states that help to overcome some of these difficulties by defining a family of states that depend only on a small number of parameters. Their use is twofold: they are used as variational ansatzes in numerical algorithms as well as providing a framework to represent a large class of exactly solvable models that are believed to represent all possible phases of matter. The present thesis investigates mathematical properties of these states thus deepening the understanding of how and why Tensor Networks are suitable for the description of quantum many-body systems.
It is believed that tensor networks can represent ground states of local Hamiltonians, but how good is this representation? This question is of fundamental importance as variational algorithms based on tensor networks can only perform well if any ground state can be approximated efficiently in such a way. While any state can be written as a tensor network state, the number of parameters needed for the description might be too large. This is not the case for one-dimensional systems: only a few parameters are required to have a good approximation of their ground states; that, in turn, allows for numerical algorithms based on tensor networks performing well. The situation in two dimensions is somewhat more complicated, but it is known that ground states of local Hamiltonians can be expressed as tensor networks with sub-exponentially many parameters. In the present thesis, we improve on these existing bounds strengthening the claim that the language of tensor networks is suitable to describe many-body systems.
Another central question is how symmetries of the system such as translational invariance, time-reversal symmetry or local unitary symmetry can be reflected in tensor networks. This question is important as systems appearing in nature might intrinsically possess certain symmetries; on one hand, understanding these symmetries simplifies the description of these systems. On the other hand, the presence of symmetries leads to the appearance of novel phases – symmetry-protected topological (SPT) order, – and tensor networks provide the right language to classify these phases. In one dimension and for

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  • Abstract

certain classes of two-dimensional tensor networks (states generated by so-called injective tensors) it is well understood how symmetries of the state can be described. A general framework, however, has yet to be developed. In the present thesis, we contribute to the development of the theory in two ways. We first investigate the question for injective tensors, and generalize the existing proof for any geometry including the hyperbolic geometry used in the AdS/CFT correspondence. Second, we introduce a class of tensor network states that include previously known examples of states exhibiting SPT order. We show how symmetries are reflected in these states thus deepening the understanding of SPT order in two dimensions.

Zusammenfassung

Stark korrelierte Systeme zeigen Pha¨nomene wie Hochtemperatursupraleitung oder den Quanten-Hall-Effekt, die mit klassischen und semiklassischen Methoden nicht erkla¨rbar sind. Da die Dimension des zugrundeliegenden Hilbertraums exponentiell mit der Gro¨ße des Systems wa¨chst, versagen viele der traditionellen Ans¨atze fu¨r derartige Systeme. Es ist daher notwendig, neuartige numerische und analytische Methoden zu entwickeln, die die Physik in diesem Bereich erkla¨ren ko¨nnen.
Tensor-Netzwerkzusta¨nde k¨onnen diese Schwierigkeiten zum Teil u¨berwinden, indem sie eine Familie von Zusta¨nden definieren, die nur von einer kleinen Anzahl von Parametern abh¨angen. Diese Zust¨ande tragen auf zwei Arten zur L¨osung des Problems bei: Erstens werden sie als Variationsansatz in numerischen Algorithmen verwendet. Zweitens bieten sie einen analytischen Zugang zu einer großen Klasse genau lo¨sbarer Modelle, von denen angenommen wird, dass sie alle mo¨glichen Materiephasen repra¨sentieren. In der vorliegenden Arbeit werden mathematische Eigenschaften dieser Zust¨ande untersucht, wodurch das Versta¨ndnis dafu¨r, wie und warum Tensor-Netzwerke fu¨r die Beschreibung von Quantensystemen geeignet sind, vertieft wird.
Zuna¨chst widmen wir uns der Frage, inwiefern Tensornetzwerke Grundzusta¨nde lokaler
Hamiltonians darstellen k¨onnen. Diese Frage ist von grundlegender Bedeutung, da Variationsalgorithmen, die auf Tensornetzwerken basieren, nur dann akkurate Ergebnisse liefern ko¨nnen, wenn der Grundzustand nicht allzu weit von der zugrundeliegenden variationellen Mannigfaltigkeit entfernt ist. Zwar kann prinzipiell jeder Quantenzustand als Tensornetzwerkstatus beschrieben werden. Jedoch ist die Anzahl der fu¨r die Beschreibung erforderlichen Parameter m¨oglicherweise extrem groß. Dies ist bei eindimensionalen Systemen nicht der Fall: Nur wenige Parameter sind erforderlich, um eine gute N¨aherung ihrer Grundzusta¨nde zu erhalten. Aufgrund dieser theoretische Grundlage kann darauf vertraut werden, dass die Ergebnisse der tensornetzwerkbasierten Algorithmen akkurat sind. Die Situation in zwei Dimensionen ist komplizierter, aber es ist bekannt, dass Grundzusta¨nde lokaler Hamiltonians als Tensornetzwerke mit subexponentiell vielen Parametern ausgedru¨ckt werden k¨onnen. In der vorliegenden Arbeit verbessern wir diese bestehenden Grenzen und verst¨arken die Behauptung, dass Tensornetzwerke geeignet ist, Vielteilchensysteme zu beschreiben.
Eine weitere zentrale Frage ist, wie Symmetrien des Systems wie Translationsinvarianz, Zeitumkehrsymmetrie oder lokale Symmetrie in Tensornetzwerken reflektiert werden ko¨nnen. Das Versta¨ndnis dieser Symmetrien vereinfacht einerseits die Beschreibung der

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  • Zusammenfassung

Systeme, in denen diese Symmetrien auftreten. Auf der anderen Seite fu¨hrt das Vorhandensein von Symmetrien zum Entstehen neuer Phasen - sogenannter “symmetry protected topological phases” (SPT) -, und Tensornetzwerke liefern die richtige Beschreibung, um diese Phasen zu klassifizieren. In einer Dimension und fu¨r bestimmte Klassen von zweidimensionalen Tensornetzwerken (Zusta¨nde, die von sogenannten injektiven Tensoren erzeugt werden) ist es gut verstanden, wie Symmetrien des physikalischen System sich in ihrer Beschreibung als Tensornetzwerk widerspiegeln. Ein allgemeiner Rahmen muss jedoch noch entwickelt werden. In der vorliegenden Arbeit tragen wir auf zweierlei Weise zur Weiterentwicklung der Theorie bei. Wir untersuchen zuna¨chst die Frage nach injektiven Tensoren und verallgemeinern den vorhandenen Beweis fu¨r jede Geometrie, einschließlich der in der AdS / CFT-Korrespondenz verwendeten hyperbolischen Geometrie. Zweitens fu¨hren wir eine Klasse von Tensornetzwerkzust¨anden ein, die bereits bekannte Beispiele fu¨r Zust¨ande mit SPT-Ordnung enthalten. Wir zeigen, wie sich Symmetrien in diesen Zusta¨nden widerspiegeln, wodurch das Versta¨ndnis der SPT-Ordnung in zwei Dimensionen vertieft wird.

List of publications

First author publications:
1. Andras Molnar, Norbert Schuch, Frank Verstraete, and J. Ignacio Cirac. “Approx-

imating Gibbs states of local Hamiltonians efficiently with projected entangled pair

states”. In: Phys. Rev. B 91.4, 045138 (4 Jan. 2015), p. 045138. arXiv: 1406.2973
2. Andras Molnar, Yimin Ge, Norbert Schuch, and J. Ignacio Cirac. “A generalization

of the injectivity condition for Projected Entangled Pair States”. In: J. Math. Phys.

59, 021902 (2018) 59.2, 021902 (June 22, 2017), p. 021902. arXiv: 1706.07329v1
3. Andras Molnar, Jos´e Garre-Rubio, David P´erez- Garc´ıa, Norbert Schuch, and J. Ig-

nacio Cirac. “Normal projected entangled pair states generating the same state”. In:

New J. Phys. 20, 113017 (Nov. 2018), p. 113017. arXiv: 1804.04964
Non-first author publications:
1. Yimin Ge, Andra´s Moln´ar, and J. Ignacio Cirac. “Rapid Adiabatic Preparation of

Injective Projected Entangled Pair States and Gibbs States”. In: Phys. Rev. Lett.

116.8 (Feb. 2016), arXiv:1508.00570. arXiv: 1508.00570
2. Ilya Kull, Andras Molnar, Erez Zohar, and J. Ignacio Cirac. “Classification of

Matrix Product States with a Local (Gauge) Symmetry”. In: Annals of Physics,

Volume 386, November 2017, Pages 199-241 386 (Aug. 1, 2017), pp. 199–241. arXiv:

1708.00362v2

3. G. Scarpa et al. “Computational complexity of PEPS zero testing”. In: ArXiv eprints (Feb. 22, 2018). arXiv: 1802.08214

4. David Sauerwein, Andras Molnar, J. Ignacio Cirac, and Barbara Kraus. “Matrix
Product States: Entanglement, symmetries, and state transformations”. In: arXiv e-prints, arXiv:1901.07448 (Jan. 2019), arXiv:1901.07448. arXiv: 1901.07448

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  • Table of contents

Contents

Abstract

vvii ix

Zusammenfassung List of publications 1 Introduction

1

37
1.1 Spin lattice models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Area law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Tensor Networks

9

2.1 Graphical notation of tensor networks . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Matrix product states . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.2 Projected Entangled Pair States . . . . . . . . . . . . . . . . . . . . 13

2.1.3 Multi-scale renormalization ansatz . . . . . . . . . . . . . . . . . . 13

2.2 Representing states as TNS . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Exact representation of states . . . . . . . . . . . . . . . . . . . . . 14 2.2.2 Approximating states . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Expectation values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 MPS: transfer matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.2 PEPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Decomposition into simple objects . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.1 Structure of TI MPS . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.2 Non-TI MPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.3 PEPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5.1 PEPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.6 Virtual symmetries and topological order . . . . . . . . . . . . . . . . . . . 23

2.6.1 G-injective MPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6.2 G-injective PEPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.7 Fundamental theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.7.1 Fundamental theorems for TI MPS . . . . . . . . . . . . . . . . . . 28

2.7.2 PEPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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2.8 Parent Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.8.1 MPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.8.2 PEPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.9 Phase classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.9.1 Gapped phases in one dimension . . . . . . . . . . . . . . . . . . . 31

2.9.2 SPT phases in one dimension . . . . . . . . . . . . . . . . . . . . . 33

3 Publications

35

3.1 Approximation of Gibbs states with PEPS . . . . . . . . . . . . . . . . . . 37 3.2 Rapid adiabatic preparation of injective PEPS and Gibbs states . . . . . . 50

3.3 Semi-injective PEPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.4 Classification of MPS with a local (gauge) symmetry . . . . . . . . . . . . 133 3.5 Normal tensors generating the same state . . . . . . . . . . . . . . . . . . . 178 3.6 Computational complexity of PEPS zero testing . . . . . . . . . . . . . . . 199

3.7 Entanglement of MPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

4 Conclusion and Outlook Bibliography

225 227 233

Acknowledgements

Chapter 1 Introduction

Condensed matter physics tries to explain how macroscopic properties of materials arise from their microscopic structures. This understanding can be used to engineer new materials such as superconductors withstanding high magnetic fields or temperatures, or semiconductors used in modern electronics. Microscopic models in condensed matter are however very hard to solve, due to the exponential scaling of parameters with the number of particles. Despite this theoretical difficulty, the macroscopic properties of many systems can be explained as a single particle interacting with an effective environment composed of the other particles, which makes the understanding of these systems relatively easy. For some physical phenomena, however this mean-field approach fails due to the presence of strong correlations. Examples of such phenomena include high-TC superconductivity [8, 9], the fractional quantum Hall effect [10] and topological insulators.
The bottleneck in the understanding of materials consists in solving the proposed firstprinciple models; therefore, simplified, effective Hamiltonians are often introduced. These effective models, despite their simplified form, are expected to capture the relevant physics, in particular the low-energy behavior of the system. Some prominent examples are the Hubbard model which is expected to capture essential physics of high-Tc superconductivity, or the Heisenberg model proposed to explain (quantum) magnetism1. Due to the crystal structure of solids, effective models are often defined on a lattice; that is, the positions of the particles are fixed at the vertices of a lattice and the model is defined on some internal degrees of freedom of the particles. The interaction is often restricted to be local, for example to nearest-neighbor interactions. In case of the Heisenberg model, the internal degree of freedom of the particles is their spin, while their positions are, for example, fixed on a one- or two-dimensional square lattice. Their interaction is described by the Hamiltonian

X

H = −J

SiSj,

(1.1)

hi,ji

where J is the strength of the interaction, Si is the spin operator acting on the spin at position i, and the summation is restricted to neighboring positions. Apart from these

1The antiferromagnetic Heisenberg model is in fact the large U/t limit of the Hubbard model.

  • 2
  • 1. Introduction

effective models, lattice models also arise as discretized versions of first-principle models such as in quantum chemistry or in lattice gauge theory.
Despite the simplicity of effective models, solving them (that is, obtaining the ground state and low energy excitations or the Gibbs state) remains challenging. The reason is that the Hilbert space used to describe the state of the system grows exponentially with the number of particles, therefore not only finding a solution, but already describing it is very hard for large system sizes. For example, describing and storing a state of 100 particles with only two degrees of freedom (qubits) is already impossible. There are a number of ways around this “curse of dimensionality” [11] such as Quantum Monte Carlo methods, variational methods, creation of exactly solvable models or quantum simulation.
Quantum Monte Carlo methods [12] build on the observation that the physically interesting quantity is not the wave function itself, but rather expectation values (and correlations) of local observables. This observation leads to sampling techniques adapted from classical stochastic modeling. While such techniques are very effective in many situations, they often suffer from the infamous sign problem making them inadequate to simulate fermions and frustrated systems.
In variational methods, one considers a set of states that are easy to store and to calculate with, and tries to find the minimum-energy state amongst them. Obviously, this method can lead to false results if the variational ansatz does not contain states from the low-energy sector of the Hamiltonian. An example for such a variational ansatz is the set of product states in the Hartree method (mean-field theory), or the use of Slater determinants in the Hartree-Fock method [13]. The latter is widely used in quantum chemistry.
The idea behind creating exactly solvable models is the following. In certain situations such as in frustrated spin systems, obtaining information about the model is very hard. One still wants to understand how global responses emerge from the microscopic description of the model. Exactly solvable models lead to such an insight by providing models with complete understanding. This, in fact, can indicate how real world models can behave and what are the important aspects of the different models.
In quantum simulation, a well-controlled quantum system is constructed to simulate the model. Such quantum systems can either serve as fully-fledged quantum computers, or as systems where despite the lack of control over every degree of freedom, some parameters can be adjusted in a way such that the system is governed by the Hamiltonian under investigation. While experimental realization of fully-fledged quantum computers is still far away, cold atoms in optical lattices [14] and trapped ions in RF traps [15] provide enough control already in current experiments for modeling interesting Hamiltonians.
In the present thesis, we investigate properties of Tensor Network States (TNS) [16] that can be thought of both as a variational ansatz for many-body Hamiltonians and as a language for creating and analyzing exactly solvable models. Tensor Network states include the one-dimensional Matrix Product States (MPS) [17, 18] ansatz, their higher dimensional generalization, Projected Entangled Pair States (PEPS) [19, 20], and the Multi-scale Renormalization Ansatz (MERA) [21] reflecting the scale invariance of critical systems. Algorithms based on MPS are extremely successful in approximating ground states of one-dimensional many-body systems. In two dimensions, algorithms based on

  • 1.1 Spin lattice models
  • 3

PEPS are more difficult to implement but have been gaining popularity in recent years. But is it theoretically justified to use PEPS as a variational ansatz? We investigate this question and improve on already existing results on how well PEPS can approximate ground states of local Hamiltonians. Due to their success in describing relevant physics, it is believed that a classification of Tensor Networks (TNs) provides us with a classification of phases. In one dimension, this has been carried out with the help of MPS, both in the presence and absence of symmetries [22, 23]. To arrive at this classification, it was necessary to understand how symmetries of the state can be reflected in the corresponding tensor networks.
In two dimensions, TNs are known to be able to represent certain topologically ordered phases. This means that for certain types of topological quantum field theories (TQFTs), they provide examples of local Hamiltonians realizing the corresponding topological order. In the presence of symmetries, they can be used to construct examples for symmetryenriched (SET) and symmetry-protected topological (SPT) phases. In the understanding of these models, it is crucial to know how symmetries are reflected in the generating tensors. It is, however, only known for concrete examples and a special class of tensors known as injective or normal tensors. We broaden the understanding of how symmetries are represented in TNS by developing new methods for deciding whether two given tensors generate the same states or not.

1.1 Spin lattice models

Before introducing Tensor Networks, let us briefly recall what spin lattice models are and what “solving” such a model means. Apart from that, we give a mathematical definition of gapped phases, and show how some of their expected properties follow from the definition. We also recall when a phase is called topologically ordered (both intrinsic and symmetryprotected).
Let us consider spins arranged on a lattice (for example, a line in one dimension, or square lattice or Kagome lattice in two dimensions). The joint state of N such ddimensional particles is described on the Hilbert space (Cd)⊗N . We consider models where

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    Simulating quantum computation by contracting tensor networks Igor Markov1 and Yaoyun Shi2 Department of Electrical Engineering and Computer Science The University of Michigan 2260 Hayward Street Ann Arbor, MI 48109-2121, USA E-mail: imarkov shiyy @eecs.umich.edu f j g Abstract The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with T gates whose underlying graph has treewidth d can be simulated deterministically in T O(1) exp[O(d)] time, which, in particular, is polynomial in T if d = O(logT). Among many implications, we show efficient simulations for quantum formulas, defined and studied by Yao (Proceedings of the 34th Annual Symposium on Foundations of Computer Science, 352–361, 1993), and for log-depth circuits whose gates apply to nearby qubits only, a natural constraint satisfied by most physical implementations. We also show that one-way quantum computation of Raussendorf and Briegel (Physical Review Letters, 86:5188– 5191, 2001), a universal quantum computation scheme with promising physical implementations, can be efficiently simulated by a randomized algorithm if its quantum resource is derived from a small-treewidth graph. Keywords: Quantum computation, computational complexity, treewidth, tensor network, classical simula- tion, one-way quantum computation. 1Supported in part by NSF 0208959, the DARPA QuIST program and the Air Force Research Laboratory. 2Supported in part by NSF 0323555, 0347078 and 0622033. 1 Introduction The recent interest in quantum circuits is motivated by several complementary considerations. Quantum information processing is rapidly becoming a reality as it allows manipulating matter at unprecedented scale.
  • Many Electrons and the Photon Field the Many-Body Structure of Nonrelativistic Quantum Electrodynamics

    Many Electrons and the Photon Field the Many-Body Structure of Nonrelativistic Quantum Electrodynamics

    Many Electrons and the Photon Field The many-body structure of nonrelativistic quantum electrodynamics vorgelegt von M. Sc. Florian Konrad Friedrich Buchholz ORCID: 0000-0002-9410-4892 an der Fakultät II – Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades DOKTOR DER NATURWISSENSCHAFTEN Dr.rer.nat. genehmigte Dissertation PROMOTIONSAUSSCHUSS: Vorsitzende: Prof. Dr. Ulrike Woggon Gutachter: Prof. Dr. Andreas Knorr Gutachter: Dr. Michael Ruggenthaler Gutachter: Prof. Dr. Angel Rubio Gutachter: Prof. Dr. Dieter Bauer Tag der wissenschaftlichen Aussprache: 24. November 2020 Berlin 2021 ACKNOWLEDGEMENTS The here presented work is the result of a long process that naturally involved many different people. They all contributed importantly to it, though in more or less explicit ways. First of all, I want to express my gratitude to Chiara, all my friends and my (German and Italian) family. Thank you for being there! Next, I want to name Michael Ruggenthaler, who did not only supervise me over many years, contribut- ing crucially to the here presented research, but also has become a dear friend. I cannot imagine how my time as a PhD candidate would have been without you! In the same breath, I want to thank Iris Theophilou, who saved me so many times from despair over non-converging codes and other difficult moments. Also without you, my PhD would not have been the same. Then, I thank Angel Rubio for his supervision and for making possible my unforgettable and formative time at the Max-Planck Institute for the Structure and Dynamics of Matter. There are few people who have the gift to make others feel so excited about physics, as Angel does.
  • Learning Phase Transition in Ising Model with Tensor-Network Born Machines

    Learning Phase Transition in Ising Model with Tensor-Network Born Machines

    Learning Phase Transition in Ising Model with Tensor-Network Born Machines Ahmadreza Azizi Khadijeh Najafi Department of Physics Department of Physics Virginia Tech Harvard & Caltech Blacksburg, VA Cambridge, MA [email protected] [email protected] Masoud Mohseni Google Ai Venice, CA [email protected] Abstract Learning underlying patterns in unlabeled data with generative models is a chal- lenging task. Inspired by the probabilistic nature of quantum physics, recently, a new generative model known as Born Machine has emerged that represents a joint probability distribution of a given data based on Born probabilities of quantum state. Leveraging on the expressibilty and training power of the tensor networks (TN), we study the capability of Born Machine in learning the patterns in the two- dimensional classical Ising configurations that are obtained from Markov Chain Monte Carlo simulations. The structure of our model is based on Projected Entan- gled Pair State (PEPS) as a two dimensional tensor network. Our results indicate that the Born Machine based on PEPS is capable of learning both magnetization and energy quantities through the phase transitions from disordered into ordered phase in 2D classical Ising model. We also compare learnability of PEPS with another popular tensor network structure Matrix Product State (MPS) and indicate that PEPS model on the 2D Ising configurations significantly outperforms the MPS model. Furthermore, we discuss that PEPS results slightly deviate from the Monte Carlo simulations in the vicinity of the critical point which is due to the emergence of long-range ordering. 1 Introduction In the past years generative models have shown a remarkable progress in learning the probability distribution of the input data and generating new samples accordingly.
  • Entanglement and Tensor Network States

    Entanglement and Tensor Network States

    17 Entanglement and Tensor Network States Jens Eisert Freie Universitat¨ Berlin Dahlem Center for Complex Quantum Systems Contents 1 Correlations and entanglement in quantum many-body systems 2 1.1 Quantum many-body systems . 2 1.2 Clustering of correlations . 4 1.3 Entanglement in ground states and area laws . 5 1.4 The notion of the ‘physical corner of Hilbert space’ . 10 2 Matrix product states 11 2.1 Preliminaries . 11 2.2 Definitions and preparations of matrix product states . 13 2.3 Computation of expectation values and numerical techniques . 17 2.4 Parent Hamiltonians, gauge freedom, geometry, and symmetries . 23 2.5 Tools in quantum information theory and quantum state tomography . 27 3 Higher-dimensional tensor network states 29 3.1 Higher-dimensional projected entangled pair states . 29 3.2 Multi-scale entanglement renormalization . 32 4 Fermionic and continuum models 34 4.1 Fermionic models . 34 4.2 Continuum models . 35 E. Pavarini, E. Koch, and U. Schollwock¨ Emergent Phenomena in Correlated Matter Modeling and Simulation Vol. 3 Forschungszentrum Julich,¨ 2013, ISBN 978-3-89336-884-6 http://www.cond-mat.de/events/correl13 17.2 Jens Eisert 1 Correlations and entanglement in quantum many-body systems 1.1 Quantum many-body systems In this chapter we will consider quantum lattice systems as they are ubiquitous in the condensed matter context or in situations that mimic condensed matter systems, as provided, say, by sys- tems of cold atoms in optical lattices. What we mean by a quantum lattice system is that we think that we have an underlying lattice structure given: some lattice that can be captured by a graph.
  • Tensor Network States: Algorithms and Applications

    Tensor Network States: Algorithms and Applications

    Tensor Network States: Algorithms and Applications 2019-2020 Time Table 12/4 (Wed) 12/5 (Thu) 12/6 (Fri) 09:00—09:30 registration registration registration 09:30—10:30 Naoki Kawashima Tao Xiang Ian McCulloch 10:30—10:50 break break break 10:50—11:30 Pan Zhang Dong-Hee Kim Katharine Hyatt 11:30—12:10 Kouichi Okunishi Tsuyoshi Obuko Ching-Yu Huang 12:10—13:30 lunch lunch lunch 13:30—14:30 Tomotoshi Nishino Chia-Min Chung Stefan Kuhn 14:30—15:10 Kenji Harada Satoshi Morita C.-J. David Lin 15:10—15:30 break break break 15:30—16:10 Poster session discussion Yoshifumi Nakamura Daisuke Kadoh 16:10—16:50 Poster session discussion banquet updated: 2019/12/02 1 Abstracts Day One: 12/4 (Wed) Morning session Chair: Ying-Jer Kao Understanding Kitaev Related Models through Tensor Networks Naoki Kawashima, ISSP, University fo Tokyo I review recent activities related to the Kitaev model and its derivatives. Especially, we devel- oped a new series of tensor network ansatzes that describes the ground state of the Kitaev model with high accuracy. The series of ansatzes suggest the essential similarity between the classical loop gas model and the Kitaev spin liquid. We also use the ansatzes for the initial state for TEBD-type optimization. Approximately contracting arbitrary tensor networks: eicient algorithms and applications in graphical models and quantum circuit simulations Pan Zhang, Institute of Theoretical Physics, Chinese Academy of Sciences Tensor networks are powerful tools in quantum many-body problems, usually defined on lattices where eicient contraction algorithms exist.
  • Categorical Tensor Network States

    Categorical Tensor Network States

    Categorical Tensor Network States Jacob D. Biamonte,1,2,3,a Stephen R. Clark2,4 and Dieter Jaksch5,4,2 1Oxford University Computing Laboratory, Parks Road Oxford, OX1 3QD, United Kingdom 2Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore 3ISI Foundation, Torino Italy 4Keble College, Parks Road, University of Oxford, Oxford OX1 3PG, United Kingdom 5Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, United Kingdom We examine the use of string diagrams and the mathematics of category theory in the description of quantum states by tensor networks. This approach lead to a unifi- cation of several ideas, as well as several results and methods that have not previously appeared in either side of the literature. Our approach enabled the development of a tensor network framework allowing a solution to the quantum decomposition prob- lem which has several appealing features. Specifically, given an n-body quantum state |ψi, we present a new and general method to factor |ψi into a tensor network of clearly defined building blocks. We use the solution to expose a previously un- known and large class of quantum states which we prove can be sampled efficiently and exactly. This general framework of categorical tensor network states, where a combination of generic and algebraically defined tensors appear, enhances the theory of tensor network states. PACS numbers: 03.65.Fd, 03.65.Ca, 03.65.Aa arXiv:1012.0531v2 [quant-ph] 17 Dec 2011 a [email protected] 2 I. INTRODUCTION Tensor network states have recently emerged from Quantum Information Science as a gen- eral method to simulate quantum systems using classical computers.
  • Quantum Quench Dynamics and Entanglement

    Quantum Quench Dynamics and Entanglement

    © 2018 by Tianci Zhou. All rights reserved. QUANTUM QUENCH DYNAMICS AND ENTANGLEMENT BY TIANCI ZHOU DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Urbana-Champaign, 2018 Urbana, Illinois Doctoral Committee: Assistant Professor Thomas Faulkner, Chair Professor Michael Stone, Director of Research Professor Peter Abbamonte Assistant Professor Lucas Wagner Abstract Quantum quench is a non-equilibrium process where the Hamiltonian is suddenly changed during the quan- tum evolution. The change can be made by spatially local perturbations (local quench) or globally switch- ing to a completely different Hamiltonian (global quench). This thesis investigates the post-quench non- equilibrium dynamics with an emphasis on the time dependence of the quantum entanglement. We inspect the scaling of entanglement entropy (EE) to learn how correlation and entanglement built up in a quench. We begin with two local quench examples. In Chap.2, we apply a local operator to the groundstate of the quantum Lifshitz model and monitor the change of the EE. We find that the entanglement grows according to the dynamical exponent z = 2 and then saturates to the scaling dimension of the perturbing operator { a value representing its strength. In Chap.3, we study the evolution after connecting two different one-dimensional critical chains at their ends. The Loschmidt echo which measures the similarity between the evolved state and the initial one decays with a power law, whose exponent is the scaling dimension of the defect (junction). Among other conclusions, we see that the local quench dynamics contain universal information of the (critical) theory.
  • Fermionic Criticality Is Shaped by Fermi Surface Topology: a Case

    Fermionic Criticality Is Shaped by Fermi Surface Topology: a Case

    Published for SISSA by Springer Received: May 6, 2020 Revised: February 20, 2021 Accepted: March 9, 2021 Published: April 15, 2021 Fermionic criticality is shaped by Fermi surface topology: a case study of the Tomonaga-Luttinger liquid JHEP04(2021)148 Anirban Mukherjee, Siddhartha Patra and Siddhartha Lal Department of Physical Sciences, Indian Institute of Science Education and Research — Kolkata, Mohanpur Campus, West Bengal — 741246, India E-mail: [email protected], [email protected], [email protected] Abstract: We perform a unitary renormalization group (URG) study of the 1D fermionic Hubbard model. The formalism generates a family of effective Hamiltonians and many- body eigenstates arranged holographically across the tensor network from UV to IR. The URG is realized as a quantum circuit, leading to the entanglement holographic mapping (EHM) tensor network description. A topological Θ-term of the projected Hilbert space of the degrees of freedom at the Fermi surface are shown to govern the nature of RG flow towards either the gapless Tomonaga-Luttinger liquid or gapped quantum liquid phases. This results in a nonperturbative version of the Berezenskii-Kosterlitz-Thouless (BKT) RG phase diagram, revealing a line of intermediate coupling stable fixed points, while the nature of RG flow around the critical point is identical to that obtained from the weak-coupling RG analysis. This coincides with a phase transition in the many-particle entanglement, as the entanglement entropy RG flow shows distinct features for the critical and gapped phases depending on the value of the topological Θ-term. We demonstrate the Ryu-Takyanagi entropy bound for the many-body eigenstates comprising the EHM network, concretizing the relation to the holographic duality principle.