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  • Algebraic complexity,

    asymptotic spectra and

    entanglement polytopes

    Jeroen Zuiddam

  • This is an updated version of my PhD dissertation at the University of Amsterdam which was originally published as ILLC Dissertation Series DS-2018-13 with ISBN 978-94-028-1175-9. This document was compiled on April 6, 2020.

    Copyright c© 2018 by Jeroen Zuiddam

    i

  • Contents

    Preface vii

    1 Introduction 3

    1.1 Matrix multiplication . . . . . . . . . . . . . . . . . . . . . . . . . 5

    1.2 The asymptotic spectrum of tensors . . . . . . . . . . . . . . . . . 6

    1.3 Higher-order CW method . . . . . . . . . . . . . . . . . . . . . . 10

    1.4 Abstract asymptotic spectra . . . . . . . . . . . . . . . . . . . . . 11

    1.5 The asymptotic spectrum of graphs . . . . . . . . . . . . . . . . . 12

    1.6 Tensor degeneration . . . . . . . . . . . . . . . . . . . . . . . . . . 14

    1.7 Combinatorial degeneration . . . . . . . . . . . . . . . . . . . . . 15

    1.8 Algebraic branching program degeneration . . . . . . . . . . . . . 16

    1.9 Organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

    2 The theory of asymptotic spectra 19

    2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

    2.2 Semirings and preorders . . . . . . . . . . . . . . . . . . . . . . . 19

    2.3 Strassen preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    2.4 Asymptotic preorders 4∼ . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 Maximal Strassen preorders . . . . . . . . . . . . . . . . . . . . . 23

    2.6 The asymptotic spectrum X(S,6) . . . . . . . . . . . . . . . . . . 25 2.7 The representation theorem . . . . . . . . . . . . . . . . . . . . . 27

    2.8 Abstract rank and subrank R,Q . . . . . . . . . . . . . . . . . . . 27

    2.9 Topological aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    2.10 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    2.11 Subsemirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

    2.12 Subsemirings generated by one element . . . . . . . . . . . . . . . 33

    2.13 Universal spectral points . . . . . . . . . . . . . . . . . . . . . . . 34

    2.14 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

    iii

  • 3 The asymptotic spectrum of graphs; Shannon capacity 37 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 The asymptotic spectrum of graphs . . . . . . . . . . . . . . . . . 39

    3.2.1 The semiring of graph isomorphism classes G . . . . . . . . 39 3.2.2 Strassen preorder via graph homomorphisms . . . . . . . . 40 3.2.3 The asymptotic spectrum of graphs X(G) . . . . . . . . . 41 3.2.4 Shannon capacity Θ . . . . . . . . . . . . . . . . . . . . . 41

    3.3 Universal spectral points . . . . . . . . . . . . . . . . . . . . . . . 43 3.3.1 Lovász theta number ϑ . . . . . . . . . . . . . . . . . . . . 43 3.3.2 Fractional graph parameters . . . . . . . . . . . . . . . . . 43

    3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

    4 The asymptotic spectrum of tensors; matrix multiplication 49 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.2 The asymptotic spectrum of tensors . . . . . . . . . . . . . . . . . 51

    4.2.1 The semiring of tensor equivalence classes T . . . . . . . . 51 4.2.2 Strassen preorder via restriction . . . . . . . . . . . . . . . 51 4.2.3 The asymptotic spectrum of tensors X(T ) . . . . . . . . . 51 4.2.4 Asymptotic rank and asymptotic subrank . . . . . . . . . 52

    4.3 Gauge points ζ(i) . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.4 Support functionals ζθ . . . . . . . . . . . . . . . . . . . . . . . . 54 4.5 Upper and lower support functionals ζθ, ζθ . . . . . . . . . . . . . 58 4.6 Asymptotic slice rank . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

    5 Tight tensors and combinatorial subrank; cap sets 67 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.2 Higher-order Coppersmith–Winograd method . . . . . . . . . . . 70

    5.2.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.2.2 Computational remarks . . . . . . . . . . . . . . . . . . . 79 5.2.3 Examples: type sets . . . . . . . . . . . . . . . . . . . . . 80

    5.3 Combinatorial degeneration method . . . . . . . . . . . . . . . . . 81 5.4 Cap sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

    5.4.1 Reduced polynomial multiplication . . . . . . . . . . . . . 83 5.4.2 Cap sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

    5.5 Graph tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

    6 Universal points in the asymptotic spectrum of tensors; entan- glement polytopes, moment polytopes 89 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.2 Schur–Weyl duality . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.3 Kronecker and Littlewood–Richardson coefficients gλµν , c

    λ µν . . . . 92

    iv

  • 6.4 Entropy inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.5 Hilbert spaces and density operators . . . . . . . . . . . . . . . . 94 6.6 Moment polytopes P(t) . . . . . . . . . . . . . . . . . . . . . . . 95

    6.6.1 General setting . . . . . . . . . . . . . . . . . . . . . . . . 95 6.6.2 Tensor spaces . . . . . . . . . . . . . . . . . . . . . . . . . 96

    6.7 Quantum functionals F θ(t) . . . . . . . . . . . . . . . . . . . . . . 97 6.8 Outer approximation . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.9 Inner approximation for free tensors . . . . . . . . . . . . . . . . . 103 6.10 Quantum functionals versus support functionals . . . . . . . . . . 104 6.11 Asymptotic slice rank . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

    7 Algebraic branching programs; approximation and nondetermi- nism 109 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.2 Definitions and basic results . . . . . . . . . . . . . . . . . . . . . 112

    7.2.1 Computational models . . . . . . . . . . . . . . . . . . . . 112 7.2.2 Complexity classes VP, VPe, VPk . . . . . . . . . . . . . 113 7.2.3 The theorem of Ben-Or and Cleve . . . . . . . . . . . . . . 114 7.2.4 Approximation closure C . . . . . . . . . . . . . . . . . . . 117 7.2.5 Nondeterminism closure N(C) . . . . . . . . . . . . . . . . 117

    7.3 Approximation closure of VP2 . . . . . . . . . . . . . . . . . . . . 118 7.4 Nondeterminism closure of VP1 . . . . . . . . . . . . . . . . . . . 121 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

    Bibliography 127

    Glossary 141

    Summary 143

    v

  • Preface

    This is an updated version of my PhD dissertation. I thank Ronald de Wolf, Sven Polak, Yinan Li, Farrokh Labib, Monique Laurent and Péter Vrana for their helpful suggestions, which have been incorporated in this version.

    Péter Vrana, Martijn Zuiddam and Māris Ozols I thank again for proofreading the draft of the dissertation. I thank Jop Briët, Dion Gijswijt, Monique Laurent, Lex Schrijver, Péter Vrana, Matthias Christandl, Maris Ōzols, Michael Walter and Bart Sevenster for helpful discussions regarding the results in Chapter 2 and Chapter 3 of this dissertation. I again thank all my coauthors for the very fruitful collaboration that lead to this dissertation: Harry Buhrman, Matthias Christandl, Péter Vrana, Jop Briët, Chris Perry, Asger Jensen, Markus Bläser, Christian Ikenmeyer, and Karl Bringmann.

    Princeton Jeroen Zuiddam September, 2018.

    vii

  • Publications

    This dissertation is primarily based on the work in the following four papers.

    [BIZ17] Karl Bringmann, Christian Ikenmeyer, and Jeroen Zuiddam. On algebraic branching programs of small width. In Ryan O’Donnell, editor, 32nd Computational Complexity Conference (CCC), 2017. https://doi.org/10.4230/LIPIcs.CCC.2017.20

    https://arxiv.org/abs/1702.05328

    Journal of the ACM, volume 65, number 5, article 32, 2018. https://doi.org/10.1145/3209663

    [CVZ16] Matthias Christandl, Péter Vrana, and Jeroen Zuiddam. Asymptotic tensor rank of graph tensors: beyond matrix multiplication. Computational complexity, 2018. http://dx.doi.org/10.1007/s00037-018-0172-8

    https://arxiv.org/abs/1609.07476

    [CVZ18] Matthias Christandl, Péter Vrana, and Jeroen Zuiddam. Universal Points in the Asymptotic Spectrum of Tensors: Extended Abstract. In Proceedings of 50th Annual ACM SIGACT Symposium on the Theory of Computing (STOC), pages 289–296, 2018. https://doi.org/10.1145/3188745.3188766

    https://arxiv.org/abs/1709.07851

    [Zui18] Jeroen Zuiddam. The asymptotic spectrum of graphs and the Shannon capacity. Combinatorica, to appear. http://arxiv.org/abs/1807.00169

    1

    https://doi.org/10.4230/LIPIcs.CCC.2017.20 https://arxiv.org/abs/1702.05328 https://doi.org/10.1145/3209663 http://dx.doi.org/10.1007/s00037-018-0172-8 https://arxiv.org/abs/1609.07476 https://doi.org/10.1145/3188745.3188766 https://arxiv.org/abs/1709.07851 http://arxiv.org/abs/1807.00169

  • Chapter 1

    Introduction

    Volker Strassen published in 1969 his famous algorithm