news on tensor network algorithms
play

News on tensor network algorithms Romn Ors Donostia International - PowerPoint PPT Presentation

Oct 27, 2023 •566 likes •918 views

News on tensor network algorithms Romn Ors Donostia International Physics Center (DIPC) December 6th 2018 S. S. Jahromi, RO, M. Kargarian, A. Langari, PRB 97, 115162 (2018) S. S. Jahromi, RO, PRB 98, 155108 (2018) S. S. Jahromi, RO,

  1. News on tensor network algorithms Román Orús Donostia International Physics Center (DIPC) December 6th 2018 S. S. Jahromi, RO, M. Kargarian, A. Langari, PRB 97, 115162 (2018) S. S. Jahromi, RO, PRB 98, 155108 (2018) S. S. Jahromi, RO, arXiv:1808.00680 P. Schmoll, S. Singh, M. Rizzi, RO, arXiv:1809.08180 A. Kshetrimayum, M. Rizzi, J. Eisert, RO, arXiv:1809.08258 P. Schmoll, A. Haller, M. Rizzi, RO, arXiv:1812.01311

  2. Applications welcomed

  3. Tens nsor ne networks ks everyw ywhe here Quantum gravity, HEP and lattice gauge theories string theory and AdS/CFT Quantum information Strongly correlated systems and computation Classical statistical mechanics Quantum simulations Entanglement and Tensor Networks Numerical tensor calculus Nuclear physics Quantum chemistry AI, Deep learning & Linguistics Materials science

  4. Outli line ne 1) iPEPS for the ruby model 2) iPEPS & VBS phases on the star lattice 3) gPEPS and the structure matrix 4) iDMRG for SU(2)-invariant chiral ladder Overview, omit many technical details

  5. Outli line ne 1) iPEPS for the ruby model 2) iPEPS & VBS phases on the star lattice 3) gPEPS and the structure matrix 4) iDMRG for SU(2)-invariant chiral ladder S. S. Jahromi, RO, M. Kargarian, A. Langari, PRB 97, 115162 (2018)

  6. <latexit sha1_base64="4ZPoby6jmn0eEuamn8SqoWsFkI4=">ACTHicbVBNSwMxEM3Wr1q/qh69BIvgocquCHoRil6KpwrWFrp1mU3TNjbZXZKsWJf9gV48ePNXePGgiGBaV9HqQODNe2+YyfMjzpS27UcrNzU9MzuXny8sLC4trxRX1y5UGEtC6yTkoWz6oChnAa1rpjltRpKC8Dlt+IOTkd64plKxMDjXw4i2BfQC1mUEtKG8IqniI7yDXRUL3GBR30wxE0ZD8v4NsWnXsb9MuzgxJUCm50DlaZGYz0BHrv89o7q6/eK5bsXtc+C9wMlBCWdW84oPbCUksaKAJB6Vajh3pdgJSM8JpWnBjRSMgA+jRloEBCKrayTiMFG8ZpoO7oTQv0HjM/pxIQCg1FL5xCtB9NamNyP+0Vqy7h+2EBVGsaUA+F3VjnWIR8niDpOUaD40AIhk5lZM+iCBaJN/wYTgTH75L7jY23UMPtsvVY6zOPJoA2ibeSgA1RBVRDdUTQHXpCL+jVurerTfr/dOas7KZdfSrcrMfMKqyZw=</latexit> <latexit sha1_base64="4ZPoby6jmn0eEuamn8SqoWsFkI4=">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</latexit> <latexit sha1_base64="4ZPoby6jmn0eEuamn8SqoWsFkI4=">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</latexit> <latexit sha1_base64="4ZPoby6jmn0eEuamn8SqoWsFkI4=">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</latexit> The he rub ruby mo model X X σ α i σ α H = − J α j α = x,y,z α − links Reproduces the low-energy sector of the topological color code model in its gapped phase (Z 2 x Z 2 topological order) E. Bombin, M. A. Martin-Delgado, PRL 97, 180501 (2006) Coarse-graining triangles leads to brickwall lattice Study with iPEPS

  7. <latexit sha1_base64="VAfDML2SnCzWpqlIkq/yMgNv+Eg=">AB/HicbZDLSgMxFIbP1Fut9Eu3QSLIAhlpgi6EYpupKsK9gLtMGTSTBuauZBkxLHUV3HjQhG3Pog738Z0OgtPZDw8f/nkJPfizmTyrK+jcLK6tr6RnGztLW9s7tn7h+0ZQIQlsk4pHoelhSzkLaUkx2o0FxYHacbX8/8zj0VkXhnUpj6gR4GDKfEay05JrlhvuATlHDTbP7EV2imtWrKqVFVoGO4cK5NV0za/+ICJQENFOJayZ1uxciZYKEY4nZb6iaQxJmM8pD2NIQ6odCbZ8lN0rJUB8iOhT6hQpv6emOBAyjTwdGeA1UguejPxP6+XKP/CmbAwThQNyfwhP+FIRWiWBowQYniqQZMBNO7IjLCAhOl8yrpEOzFLy9Du1a1Nd+eVepXeRxFOIQjOAEbzqEON9CEFhBI4Rle4c14Ml6Md+Nj3low8pky/Cnj8wflfZJU</latexit> <latexit sha1_base64="VAfDML2SnCzWpqlIkq/yMgNv+Eg=">AB/HicbZDLSgMxFIbP1Fut9Eu3QSLIAhlpgi6EYpupKsK9gLtMGTSTBuauZBkxLHUV3HjQhG3Pog738Z0OgtPZDw8f/nkJPfizmTyrK+jcLK6tr6RnGztLW9s7tn7h+0ZQIQlsk4pHoelhSzkLaUkx2o0FxYHacbX8/8zj0VkXhnUpj6gR4GDKfEay05JrlhvuATlHDTbP7EV2imtWrKqVFVoGO4cK5NV0za/+ICJQENFOJayZ1uxciZYKEY4nZb6iaQxJmM8pD2NIQ6odCbZ8lN0rJUB8iOhT6hQpv6emOBAyjTwdGeA1UguejPxP6+XKP/CmbAwThQNyfwhP+FIRWiWBowQYniqQZMBNO7IjLCAhOl8yrpEOzFLy9Du1a1Nd+eVepXeRxFOIQjOAEbzqEON9CEFhBI4Rle4c14Ml6Md+Nj3low8pky/Cnj8wflfZJU</latexit> <latexit sha1_base64="VAfDML2SnCzWpqlIkq/yMgNv+Eg=">AB/HicbZDLSgMxFIbP1Fut9Eu3QSLIAhlpgi6EYpupKsK9gLtMGTSTBuauZBkxLHUV3HjQhG3Pog738Z0OgtPZDw8f/nkJPfizmTyrK+jcLK6tr6RnGztLW9s7tn7h+0ZQIQlsk4pHoelhSzkLaUkx2o0FxYHacbX8/8zj0VkXhnUpj6gR4GDKfEay05JrlhvuATlHDTbP7EV2imtWrKqVFVoGO4cK5NV0za/+ICJQENFOJayZ1uxciZYKEY4nZb6iaQxJmM8pD2NIQ6odCbZ8lN0rJUB8iOhT6hQpv6emOBAyjTwdGeA1UguejPxP6+XKP/CmbAwThQNyfwhP+FIRWiWBowQYniqQZMBNO7IjLCAhOl8yrpEOzFLy9Du1a1Nd+eVepXeRxFOIQjOAEbzqEON9CEFhBI4Rle4c14Ml6Md+Nj3low8pky/Cnj8wflfZJU</latexit> <latexit sha1_base64="VAfDML2SnCzWpqlIkq/yMgNv+Eg=">AB/HicbZDLSgMxFIbP1Fut9Eu3QSLIAhlpgi6EYpupKsK9gLtMGTSTBuauZBkxLHUV3HjQhG3Pog738Z0OgtPZDw8f/nkJPfizmTyrK+jcLK6tr6RnGztLW9s7tn7h+0ZQIQlsk4pHoelhSzkLaUkx2o0FxYHacbX8/8zj0VkXhnUpj6gR4GDKfEay05JrlhvuATlHDTbP7EV2imtWrKqVFVoGO4cK5NV0za/+ICJQENFOJayZ1uxciZYKEY4nZb6iaQxJmM8pD2NIQ6odCbZ8lN0rJUB8iOhT6hQpv6emOBAyjTwdGeA1UguejPxP6+XKP/CmbAwThQNyfwhP+FIRWiWBowQYniqQZMBNO7IjLCAhOl8yrpEOzFLy9Du1a1Nd+eVepXeRxFOIQjOAEbzqEON9CEFhBI4Rle4c14Ml6Md+Nj3low8pky/Cnj8wflfZJU</latexit> Ene nergies a and nd pha hase di diag agram ram Ground state energy Multicritical point −1.05 −0.55 J c =(0.8, 0.8, 0.4) J=(1, 1, 1) −1.06 −0.56 iPEPS FU iPEPS FU Comparison to exact diag. ED (N=24) −1.07 ED (N=24) −0.57 −1.08 for 24 sites. −0.58 −1.09 ε 0 ε 0 −1.1 −0.59 Good convergence. −1.11 −0.6 Multicritical point challenging. −1.12 −0.61 −1.13 −1.14 −0.62 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 1/D 1/D Phase diagram 3 phases separated by 2nd order QPTs. J x + J y + J z = 2 QPTs meet at a multicritical point. A 1 phase: gapped, Z 2 x Z 2 TO. A 2 & A 3 phases: gapless. Gapped by a perturbation, colored Ising anyons. S. S. Jahromi, M. Kargarian, S. F. Masoudi, A. Langari, PRB 94, 125145 (2016)

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Renormalization of Tensor Network States II. RG of Tensor Network States Tao Xiang Institute of

Renormalization of Tensor Network States II. RG of Tensor Network States Tao Xiang Institute of

Renormalization of Tensor Network States II. RG of Tensor Network States Tao Xiang Institute of Physics Chinese Academy of Sciences txiang@iphy.ac.cn Tensor-Network Ansatz of the ground state wavefunction 1D: Matrix Product State (MPS) : m

603 views • 57 slides
Parallel Numerical Algorithms Chapter 7 Differential Equations Section 7.5 Tensor

Parallel Numerical Algorithms Chapter 7 Differential Equations Section 7.5 Tensor

Tensor Algebra Tensor Decompositions Fast Algorithms Parallel Numerical Algorithms Chapter 7 Differential Equations Section 7.5 Tensor Analysis Edgar Solomonik Department of Computer Science University of Illinois at Urbana-Champaign

722 views • 26 slides
and You Tensor network methods Matrix product states (MPS) Projected Entangled Pair States

and You Tensor network methods Matrix product states (MPS) Projected Entangled Pair States

Tensor Networks and You Nikko Pomata How do you network tensors? Tensors: a review The tensor-network notation T ENSOR N ETWORKS Tensor network examples and You Tensor network methods Matrix product states (MPS) Projected Entangled Pair

642 views • 24 slides
Scalable Tensor Computations with Cyclops and Faster Algorithms for Alternating Least Squares

Scalable Tensor Computations with Cyclops and Faster Algorithms for Alternating Least Squares

Scalable Tensor Computations with Cyclops and Faster Algorithms for Alternating Least Squares Edgar Solomonik Department of Computer Science University of Illinois at Urbana-Champaign Invited Workshop on Compiler Techniques for Sparse Tensor

222 views • 9 slides
Tensor Network Approach to Real-Time Path Integral Shinji Takeda (Kanazawa U.) Frontiers in

Tensor Network Approach to Real-Time Path Integral Shinji Takeda (Kanazawa U.) Frontiers in

Tensor Network Approach to Real-Time Path Integral Shinji Takeda (Kanazawa U.) Frontiers in Lattice QCD and related topics 2019.04.15-26 @YITP Contents Introduction to tensor network approach Why/Whats tensor network

1.13k views • 79 slides
Sparse Tensor Factorization: Algorithms, Data Structures, and Challenges Shaden Smith &amp;

Sparse Tensor Factorization: Algorithms, Data Structures, and Challenges Shaden Smith &amp;

Sparse Tensor Factorization: Algorithms, Data Structures, and Challenges Shaden Smith &amp; George Karypis University of Minnesota Department of Computer Science &amp; Engineering shaden@cs.umn.edu Sparse Tensor Factorization: Algorithms,

844 views • 80 slides
Models through Tensor Networks 2019.12.04 Naoki KAWASHIMA (ISSP) Tensor Network (TN) (1)

Models through Tensor Networks 2019.12.04 Naoki KAWASHIMA (ISSP) Tensor Network (TN) (1)

TNSAA2019-2020 2019.12.04-06 Understanding Kitaev-Related Models through Tensor Networks 2019.12.04 Naoki KAWASHIMA (ISSP) Tensor Network (TN) (1) Classical statistical mechanical models are TNs --- corner

282 views • 25 slides
Tensor Network Representation for Machine Learning - Recent Advances and Perspectives Qibin ZHAO

Tensor Network Representation for Machine Learning - Recent Advances and Perspectives Qibin ZHAO

Tensor Network Representation for Machine Learning - Recent Advances and Perspectives Qibin ZHAO Tensor Learning Unit RIKEN AIP AIP Symposium (Mar. 19, 2019) Tensor Learning Unit - Members Postdoctoral Researchers (2) Ming Hou, Chao Li

543 views • 28 slides
Tensor For F , F Sh( X ) we have F F Sh( X ). Since tensor commutes with

Tensor For F , F Sh( X ) we have F F Sh( X ). Since tensor commutes with

Tensor For F , F Sh( X ) we have F F Sh( X ). Since tensor commutes with taking stalk, the non-exactness comes from that F x might not torsion-free, and this can be taken care by a simple resolution 0 P P F

1.85k views • 7 slides
5 Network Layer Network Layer Network Layer Network Layer Example: Choosing among multiple ASes

5 Network Layer Network Layer Network Layer Network Layer Example: Choosing among multiple ASes

Network Layer Network Layer Network Layer Network Layer Network Layer Comparison of LS and DV algorithms Message complexity Robustness: what happens 1 Introduction 5 Routing algorithms if router malfunctions? LS: with n nodes, E

325 views • 4 slides
? iPEPS 0 0 . 189(2) 0 . 217(4) 0 . 25 2D tensor network ansatz I. Niesen and P. Corboz,

? iPEPS 0 0 . 189(2) 0 . 217(4) 0 . 25 2D tensor network ansatz I. Niesen and P. Corboz,

2D tensor network study of the S=1 bilinear-biquadratic Heisenberg model Philippe Corboz, Institute for Theoretical Physics, University of Amsterdam AF phase Haldane phase 3-SL 120 phase ? iPEPS 0 0 . 189(2) 0 . 217(4) 0 . 25 2D

758 views • 51 slides
8. Tensor Field Visualization Tensor: extension of concept of scalar and vector Tensor data

8. Tensor Field Visualization Tensor: extension of concept of scalar and vector Tensor data

8. Tensor Field Visualization Tensor: extension of concept of scalar and vector Tensor data for a tensor of level k is given by t i1,i2,,ik (x 1 ,,x n ) Second-order tensor often represented by matrix Examples:

292 views • 15 slides
An Exploration of Optimization Algorithms for High Performance Tensor Completion Shaden Smith 1

An Exploration of Optimization Algorithms for High Performance Tensor Completion Shaden Smith 1

An Exploration of Optimization Algorithms for High Performance Tensor Completion Shaden Smith 1 , Jongsoo Park 2 , and George Karypis 1 1 Department of Computer Science &amp; Engineering, University of Minnesota 2 Parallel Computing Lab, Intel

414 views • 39 slides
Breaking the News: Extracting the Sparse Citation Network Backbone of Online News Articles

Breaking the News: Extracting the Sparse Citation Network Backbone of Online News Articles

Breaking the News: Extracting the Sparse Citation Network Backbone of Online News Articles Andreas Spitz and Michael Gertz Heidelberg University Institute of Computer Science Database Systems Research Group http://dbs.ifi.uni-heidelberg.de

285 views • 27 slides
Breaking the News: Extracting the Sparse Citation Network Backbone of Online News Articles

Breaking the News: Extracting the Sparse Citation Network Backbone of Online News Articles

Breaking the News: Extracting the Sparse Citation Network Backbone of Online News Articles Andreas Spitz and Michael Gertz Heidelberg University Institute of Computer Science Database Systems Research Group http://dbs.ifi.uni-heidelberg.de

855 views • 53 slides
Our News, Your Branding WINNER OF THE 2017 EDWARD R MURROW AWARD FOR HARD NEWS REPORTING

Our News, Your Branding WINNER OF THE 2017 EDWARD R MURROW AWARD FOR HARD NEWS REPORTING

Our News, Your Branding WINNER OF THE 2017 EDWARD R MURROW AWARD FOR HARD NEWS REPORTING Westwood One News is a next-generation news product that is changing the nature of network news. Westwood One News Award Winning Fully

644 views • 12 slides
DevOps Pay Raise Quantifying your value to move up the ladder Tom Levey Tech Evangelist @tlevey

DevOps Pay Raise Quantifying your value to move up the ladder Tom Levey Tech Evangelist @tlevey

DevOps Pay Raise Quantifying your value to move up the ladder Tom Levey Tech Evangelist @tlevey DevOps ? Developers working together with Operations to get things done faster in an automated and repeatable way DevOps Success ? 2005

773 views • 46 slides
HALI: Hierarchical Adversarially Learned Inference Negar Rostamzadeh ACM Webinar January 18,

HALI: Hierarchical Adversarially Learned Inference Negar Rostamzadeh ACM Webinar January 18,

HALI: Hierarchical Adversarially Learned Inference Negar Rostamzadeh ACM Webinar January 18, 2018 Negar Rostamzadeh HALI January 18, 2018 1 / 43 Hierarchical Adversarially Learned Inference Mohamed Ishmael Belghazi, Sai Rajeswar, Olivier

468 views • 43 slides
7.2 inverse Laplace Transforms, and application to DEs a lesson for MATH F302 Differential

7.2 inverse Laplace Transforms, and application to DEs a lesson for MATH F302 Differential

7.2 inverse Laplace Transforms, and application to DEs a lesson for MATH F302 Differential Equations Ed Bueler, Dept. of Mathematics and Statistics, UAF March 22, 2019 for textbook: D. Zill, A First Course in Differential Equations with

215 views • 18 slides
Improved Security Bounds for Generalized Feistel Networks Yaobin Shen 1 Chun Guo 2 Lei Wang 1 1

Improved Security Bounds for Generalized Feistel Networks Yaobin Shen 1 Chun Guo 2 Lei Wang 1 1

Feistel Networks Our Contributions Security Proofs Conclusion Improved Security Bounds for Generalized Feistel Networks Yaobin Shen 1 Chun Guo 2 Lei Wang 1 1 Shanghai Jiao Tong University 2 Shandong University November 13, FSE 2020 Yaobin

407 views • 22 slides
Performance Optimization 2 Lab Schedule Activities Assignments Due Today Lab 5 Due

Performance Optimization 2 Lab Schedule Activities Assignments Due Today Lab 5 Due

Computer Systems and Networks ECPE 170 Jeff Shafer University of the Pacific Performance Optimization 2 Lab Schedule Activities Assignments Due Today Lab 5 Due by Feb 26 th 5:00am Discussion on Performance

447 views • 41 slides
CHIRAL TOPOLOGICAL, AND GAPLESS NON-FERMI-LIQUID PHASE ANDREAS W.W. LUDWIG (UC-Santa Barbara)

CHIRAL TOPOLOGICAL, AND GAPLESS NON-FERMI-LIQUID PHASE ANDREAS W.W. LUDWIG (UC-Santa Barbara)

SPIN-LIQUIDS ON THE KAGOME LATTICE: CHIRAL TOPOLOGICAL, AND GAPLESS NON-FERMI-LIQUID PHASE ANDREAS W.W. LUDWIG (UC-Santa Barbara) work done in collaboration with: Bela Bauer (Microsoft Station-Q, Santa Barbara) Simon Trebst (Univ. of

666 views • 52 slides
8 LI 8 LIFES ESTYLE LE T TIPS F FOR A A HEA HEALTHY HY BRAIN N Beverly A A. . Jackey

8 LI 8 LIFES ESTYLE LE T TIPS F FOR A A HEA HEALTHY HY BRAIN N Beverly A A. . Jackey

8 LI 8 LIFES ESTYLE LE T TIPS F FOR A A HEA HEALTHY HY BRAIN N Beverly A A. . Jackey MS MS, , RDN Fam amily an and Co d Consumer Sc Scie ience E Edu ducator Univ iversit ity o of Marylan and E d Extensio ion Image

398 views • 20 slides
Urban Computing Dr. Mitra Baratchi 28 September 2020 Leiden Institute of Advanced Computer

Urban Computing Dr. Mitra Baratchi 28 September 2020 Leiden Institute of Advanced Computer

Urban Computing Dr. Mitra Baratchi 28 September 2020 Leiden Institute of Advanced Computer Science - Leiden University 1 Fourth Session: Urban Computing - Processing spatiotemporal data 2 Table of Contents 1. Preliminaries What is

669 views • 65 slides

More recommend