DOCSLIB.ORG

Tensor Networks, MERA & 2D MERA ✦ Classify Tensor Network Ansatz According to Its Entanglement Scaling

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Tensor Networks, MERA & 2D MERA ✦ Classify Tensor Network Ansatz According to Its Entanglement Scaling Lecture 1: tensor network states (MPS, PEPS & iPEPS, Tree TN, MERA, 2D MERA) Philippe Corboz, Institute for Theoretical Physics, University of Amsterdam i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11i12 i13 i14 i15i16 i17 i18 Outline ‣ Lecture 1: tensor network states ✦ Main idea of a tensor network ansatz & area law of the entanglement entropy ✦ MPS, PEPS & iPEPS, Tree tensor networks, MERA & 2D MERA ✦ Classify tensor network ansatz according to its entanglement scaling ‣ Lecture II: tensor network algorithms (iPEPS) ✦ Contraction & Optimization ‣ Lecture III: Fermionic tensor networks ✦ Formalism & applications to the 2D Hubbard model ✦ Other recent progress Motivation: Strongly correlated quantum many-body systems High-Tc Quantum magnetism / Novel phases with superconductivity spin liquids ultra-cold atoms Challenging!tech-faq.com Typically: • No exact analytical solution Accurate and efficient • Mean-field / perturbation theory fails numerical simulations • Exact diagonalization: O(exp(N)) are essential! Quantum Monte Carlo • Main idea: Statistical sampling of the exponentially large configuration space • Computational cost is polynomial in N and not exponential Very powerful for many spin and bosonic systems Quantum Monte Carlo • Main idea: Statistical sampling of the exponentially large configuration space • Computational cost is polynomial in N and not exponential Very powerful for many spin and bosonic systems Example: The Heisenberg model . Sandvik & Evertz, PRB 82 (2010): . H = SiSj system sizes up to 256x256 i,j 65536 ⇥ Hilbert space: 2 Ground state sublattice magn. m =0.30743(1) has Néel order . Quantum Monte Carlo • Main idea: Statistical sampling of the exponentially large configuration space • Computational cost is polynomial in N and not exponential Very powerful for many spin and bosonic systems BUT Quantum Monte Carlo & the negative sign problem Bosons Fermions (e.g. 4He) (e.g electrons) Ψ (x ,x )=Ψ (x ,x ) Ψ (x ,x )= Ψ (x ,x ) B 1 2 B 2 1 F 1 2 − F 2 1 symmetric! antisymmetric! this leads to the infamous ✓ negative sign problem t (poly(N/T )) t (exp(N/T )) sim ∼ O sim ∼ O cannot solve large systems at low temperature! Strongly correlated fermionic systems ˆ 2D Hubbard model H = t cˆi†σcˆjσ + U nˆi nˆi − ↑ ⇥ i,j ,σ i • High-temperature superconductors ⇤⌅ Hopping between On-site repulsion between nearest-neighbor sites electrons with opposite spin Is it the relevant model of high-temperature superconductors? Quantum Monte Carlo & the negative sign problem Non-frustrated Frustrated spin systems spin systems frustrated ? all bonds satisfied not all bonds satisfied this leads to the infamous ✓ negative sign problem t (poly(N/T )) t (exp(N/T )) sim ∼ O sim ∼ O cannot solve large systems at low temperature! To make prog ress in strongly correlated systems it is essential to negative develop new accurate numerical methods! sign problem • DMFT / DCA • Diagrammatic Monte Carlo • Tensor network algorithms • Fixed-node Monte Carlo • Series expansion • Density Matrix Embedding Theory • Variational Monte Carlo • Functional renormalization group • Coupled-cluster methods • ... Overview: tensor networks in 1D and 2D 1D MPS 1D MERA Matrix-product state Multi-scale entanglement renormalization ansatz and more 1 2 3 4 5 6 7 8 ‣ 1D tree tensor Underlying ansatz of the network density-matrix renormalization ‣ correlator product states group (DMRG) method i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11i12 i13 i14 i15i16 i17 i18 ‣ ... 2D PEPS 2D MERA and more projected entangled-pair state ‣ Entangled- plaquette states ‣ 2D tree tensor network ‣ String-bond states ‣ ... Diagrammatic notation l c i v i M j i A j i B j ... k k number vector vi matrix Mij Aijk Bijkl rank-0 tensor rank-1 tensor rank-2 tensor rank-3 tensor rank-4 tensor ★ We don’t need to write down formulas with tensors with many indices! i j Example 1: M v = i u Mijvj = ui j X ★ Connected lines: sum over corresponding indices! Diagrammatic notation i j Example 2: u M v = c ★ sum over all connected indices: contraction of a tensor network uiMijvj = c ij X B j Example 3: i = T A C ★ The rank of the k resulting tensor u v u v corresponds to the number of open legs in the network AuikBijCvjk = Tuv Xijk Diagrammatic notation ★ Hard to write down with all indices... ★ We know the result is going to be a number Introduction to tensor networks ➡ Aim: Efficient representation of quantum many-body states V : local Hilbert space Lattice with .... e.g. , N sites 1 2 3 4 5 N-1 N {| ⇧ | ⇥⇧} Full Hilbert dimension 2N .... space V⊗V⊗V⊗V⊗V⊗ ⊗V⊗V grows exponentially with N Hamiltonian Hˆ = hˆij sum of local terms ij ⇥ Represent the Ψ = Ψ i i i ground state | ⇤ i1i2...iN | 1 ⇥ 2 ⇥ ···⇥ N ⇤ i i ...i 1 2 N i , k ⇤ {↑ ⇥} 2N coefficients Complexity ~exp(N) many numbers inefficient! Tensor network ansatz for a wave function 2 basis states per site: , {| "i | #i} Lattice: 26 basis states 1 2 3 4 5 6 26 coefficients State: Ψ = Ψ i i i i i i | ⇥ i1i2i3i4i5i6 | 1 ⊗ 2 ⊗ 3 ⊗ 4 ⊗ 5 ⊗ 6⇥ i i i i i i 1 23 4 5 6 Tensor/multidimensional array Tensor network: matrix product state (MPS) Why is athisb c d e Big tensor i1i2i3i4i5i6 A B C D E F D bond dimension i1 i2 i3 i4 i5 i6 i1 i2 i3 i4 i5 possible??i6 Aa BabCbcDcdEdeF e = ˜ Ψi1i2i3i4i5i6 i1 i2 i3 i4 i5 i6 i1i2i3i4i5i6 ⇡ abcdeX Efficient exp(N) many numbers vs poly(D ,N) numbers representation! “Corner” of the Hilbert space Ground states (local H) ★ GS of local H’s are less entangled than a Hilbert random state in the Hilbert space space ★ Area law of the entanglement entropy Splitting in the middle Left side Right side Big tensor i1i2i3i4i5i6 i1 i2 i3 i4 i5 i6 i1 i2 i3 i4 i5 i6 Singular value decomposition U s V† = UsV † skk 0 l r l, r 1,...,M l r 2 { } ≥ M =2N/2 diagonal matrix! lr = UlkskkVrk⇤ Xk = l r = UlkskkV ⇤ l r lr rk| i| i | i | i| i lr k Xlr X X Schmidt = skk uk vk | i| i decomposition Xk How many relevant singular values? M how many non-zero = s u v U s V† | i kk| ki| ki singular values? Xk l r ★ Special cases: s11 =1,skk = 0 for k>1 =1u v Product state | i | 1i| 1i 1 1 s11 = ,s22 = ,skk = 0 for k>2 p2 p2 1 1 = u1 v1 + u2 v2 Entangled state | i p2| i| i p2| i| i 1 Maximally skk = , for all k pM entangled state How many relevant singular values? bond dimension D = χ M how many relevant = s u v U s V† | i kk| ki| ki singular values? Xk skk l r χ ˜ = s u v | i⇡| i kk| ki| ki Ground state (local H) Xk random state: keeping the χ largest all singular values important singular values minimizes the error ˜ k ||| i| i|| χ M KEY IDEA OF relevant singular values truncate this part DMRG! Reduced density matrix M A B = s u v | i kk| ki| ki Xk ★ Reduced density matrix of left side: describes system on the left side 2 ⇢ =tr [⇢]=tr [ ]= λ u u λk = s probability A B B | ih | k| kih k| kk Xk ★ Entanglement entropy: S(A)= tr[⇢ log ⇢ ]= λ log λ − A A − k k Xk ‣ Product state: S(A)= 1 log 1 = 0 ... − 1 1 ‣ Maximally entangled state: S(A)= log = logM − M M Xk How large is S in a ground state? How does it scale with system size? . Area law of the entanglement entropy E . 1D 2D A E A E . L L . # relevant states Entanglement entropy S(A)= tr[⇥ log ⇥ ]= λ log λ − A A − i i χ exp(S) i ⇠ General (random) state Ground state (local Hamiltonian) d d 1 S(L) L (volume) S(L) L − (area law) ⇠ ⇠ Critical ground states: 1D S(L)=const χ = const (all in 1D but not all in 2D) 2D S(L) αL ⇥ exp(αL) 1D S(L) log(L) ∼ ∼ ⇠ 2D S(L) L log(L) ⇠ MPS & PEPS 1D MPS Matrix-product state Bond dimension D 1 2 3 4 5 6 7 8 Physical indices (lattices sites) S. R. White, PRL 69, 2863 (1992) Fannes et al., CMP 144, 443 (1992) Östlund, Rommer, PRL 75, 3537 (1995) ✓ Reproduces area-law in 1D S(L)=const MPS & PEPS 1D MPS Matrix-product state Bond dimension D ➡ One bond can contribute at most log(D) to the A E entanglement entropy L rank(⇢ ) D S(A) log(D)=const A ✓ Reproduces area-law in 1D S(L)=const MPS & PEPS 1D 2D MPS can we use Matrix-product state an MPS? Bond dimension D 1 2 3 4 5 6 7 8 Physical indices (lattices sites) L S. R. White, PRL 69, 2863 (1992) Fannes et al., CMP 144, 443 (1992) Östlund, Rommer, PRL 75, 3537 (1995) !!! Area-law in 2D !!! ✓ Reproduces area-law in 1D S(L) L ∼ S(L)=const D exp(L) ⇠ MPS & PEPS 1D 2D PEPS (TPS) MPS projected entangled-pair state Matrix-product state (tensor product state) Bond dimension D Bond dimension D 1 2 3 4 5 6 7 8 Physical indices (lattices sites) S. R. White, PRL 69, 2863 (1992) Fannes et al., CMP 144, 443 (1992) Östlund, Rommer, PRL 75, 3537 (1995) F. Verstraete, J. I. Cirac, cond-mat/0407066 Nishio, Maeshima, Gendiar, Nishino, cond-mat/0401115 ✓ Reproduces area-law in 1D ✓ Reproduces area-law in 2D S(L)=const S(L) L ∼ PEPS: Area law one “thick” bond of dimension DL DL A B ... ... L S(A) L log D L ⇠ each cut auxiliary bond can contribute (at most) log D to the entanglement entropy The number of cuts scales ✓ Reproduces area-law in 2D with the cut length S(L) L ∼ MPS & PEPS 1D 2D PEPS (TPS) MPS projected entangled-pair state Matrix-product state (tensor product state) Bond dimension D Bond dimension D 1 2 3 4 5 6 7 8 Physical indices (lattices sites) S.
Load more

Recommended publications
  • Tensor Network Methods in Many-Body Physics Andras Molnar
    Tensor Network Methods in Many-body physics Andras Molnar Ludwig-Maximilians-Universit¨atM¨unchen Max-Planck-Institut f¨urQuanutenoptik M¨unchen2019 Tensor Network Methods in Many-body physics Andras Molnar Dissertation an der Fakult¨atf¨urPhysik der Ludwig{Maximilians{Universit¨at M¨unchen vorgelegt von Andras Molnar aus Budapest M¨unchen, den 25/02/2019 Erstgutachter: Prof. Dr. Jan von Delft Zweitgutachter: Prof. Dr. J. Ignacio Cirac Tag der m¨undlichen Pr¨ufung: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.
    [Show full text]
  • Tensor Network State Methods and Applications for Strongly Correlated Quantum Many-Body Systems
    Bildquelle: Universität Innsbruck Tensor Network State Methods and Applications for Strongly Correlated Quantum Many-Body Systems Dissertation zur Erlangung des akademischen Grades Doctor of Philosophy (Ph. D.) eingereicht von Michael Rader, M. Sc. betreut durch Univ.-Prof. Dr. Andreas M. Läuchli an der Fakultät für Mathematik, Informatik und Physik der Universität Innsbruck April 2020 Zusammenfassung Quanten-Vielteilchen-Systeme sind faszinierend: Aufgrund starker Korrelationen, die in die- sen Systemen entstehen können, sind sie für eine Vielzahl an Phänomenen verantwortlich, darunter Hochtemperatur-Supraleitung, den fraktionalen Quanten-Hall-Effekt und Quanten- Spin-Flüssigkeiten. Die numerische Behandlung stark korrelierter Systeme ist aufgrund ih- rer Vielteilchen-Natur und der Hilbertraum-Dimension, die exponentiell mit der Systemgröße wächst, extrem herausfordernd. Tensor-Netzwerk-Zustände sind eine umfangreiche Familie von variationellen Wellenfunktionen, die in der Physik der kondensierten Materie verwendet werden, um dieser Herausforderung zu begegnen. Das allgemeine Ziel dieser Dissertation ist es, Tensor-Netzwerk-Algorithmen auf dem neuesten Stand der Technik für ein- und zweidi- mensionale Systeme zu implementieren, diese sowohl konzeptionell als auch auf technischer Ebene zu verbessern und auf konkrete physikalische Systeme anzuwenden. In dieser Dissertation wird der Tensor-Netzwerk-Formalismus eingeführt und eine ausführ- liche Anleitung zu rechnergestützten Techniken gegeben. Besonderes Augenmerk wird dabei auf die Implementierung
    [Show full text]
  • Continuous Tensor Network States for Quantum Fields
    PHYSICAL REVIEW X 9, 021040 (2019) Featured in Physics Continuous Tensor Network States for Quantum Fields † Antoine Tilloy* and J. Ignacio Cirac Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany (Received 3 August 2018; revised manuscript received 12 February 2019; published 28 May 2019) We introduce a new class of states for bosonic quantum fields which extend tensor network states to the continuum and generalize continuous matrix product states to spatial dimensions d ≥ 2.By construction, they are Euclidean invariant and are genuine continuum limits of discrete tensor network states. Admitting both a functional integral and an operator representation, they share the important properties of their discrete counterparts: expressiveness, invariance under gauge transformations, simple rescaling flow, and compact expressions for the N-point functions of local observables. While we discuss mostly the continuous tensor network states extending projected entangled-pair states, we propose a generalization bearing similarities with the continuum multiscale entanglement renormal- ization ansatz. DOI: 10.1103/PhysRevX.9.021040 Subject Areas: Particles and Fields, Quantum Physics, Strongly Correlated Materials I. INTRODUCTION and have helped describe and classify their physical proper- ties. By design, their entanglement obeys the area law Tensor network states (TNSs) provide an efficient para- [17–19], which is a fundamental property of low-energy metrization of physically relevant many-body wave func- states of systems with local interactions. They enable a tions on the lattice [1,2]. Obtained from a contraction of succinct classification of symmetry-protected [20–23] and low-rank tensors on so-called virtual indices, they eco- topological phases of matter [24,25].
    [Show full text]
  • Numerical Continuum Tensor Networks in Two Dimensions
    PHYSICAL REVIEW RESEARCH 3, 023057 (2021) Numerical continuum tensor networks in two dimensions Reza Haghshenas ,* Zhi-Hao Cui , and Garnet Kin-Lic Chan† Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA (Received 31 August 2020; accepted 31 March 2021; published 19 April 2021) We describe the use of tensor networks to numerically determine wave functions of interacting two- dimensional fermionic models in the continuum limit. We use two different tensor network states: one based on the numerical continuum limit of fermionic projected entangled pair states obtained via a tensor network for- mulation of multigrid and another based on the combination of the fermionic projected entangled pair state with layers of isometric coarse-graining transformations. We first benchmark our approach on the two-dimensional free Fermi gas then proceed to study the two-dimensional interacting Fermi gas with an attractive interaction in the unitary limit, using tensor networks on grids with up to 1000 sites. DOI: 10.1103/PhysRevResearch.3.023057 I. INTRODUCTION and there have been many developments to extend the range of the techniques, for example to long-range Hamiltoni- Understanding the collective behavior of quantum many- ans [15–17], thermal states [18], and real-time dynamics [19]. body systems is a central theme in physics. While it is often Also, much work has been devoted to improving the numeri- discussed using lattice models, there are systems where a cal efficiency and stability of PEPS computations [20–24]. continuum description is essential. One such case is found Formulating tensor network states and the associated algo- in superfluids [1], where recent progress in precise experi- rithms in the continuum remains a challenge.
    [Show full text]
  • Simulating Quantum Computation by Contracting Tensor Networks Abstract
    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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]