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 Lov´asztheta number # .................... 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 θ 4.5 Upper and lower support functionals ζ ; 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 λ 6.3 Kronecker and Littlewood{Richardson coefficients gλµν, cµν .... 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 P´eterVrana for their helpful suggestions, which have been incorporated in this version. P´eterVrana, Martijn Zuiddam and M¯arisOzols I thank again for proofreading the draft of the dissertation. I thank Jop Bri¨et,Dion Gijswijt, Monique Laurent, Lex Schrijver, P´eterVrana, Matthias Christandl, Maris Ozols,¯ 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, P´eterVrana, Jop Bri¨et,Chris Perry, Asger Jensen, Markus Bl¨aser,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, P´eterVrana, 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, P´eterVrana, 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 Chapter 1 Introduction Volker Strassen published in 1969 his famous algorithm for multiplying any two n × n matrices using only O(n2:81) rather than O(n3) arithmetical opera- tions [Str69]. His discovery marked the beginning of a still ongoing line of research in the field of algebraic complexity theory; a line of research that by now touches several fields of mathematics, including algebraic geometry, representation theory, (quantum) information theory and combinatorics. This dissertation is inspired by and contributes to this line of research. No further progress followed for almost 10 years after Strassen's discovery, despite the fact that \many scientists understood that discovery as a signal to attack the problem and to push the exponent further down" [Pan84]. Then in 1978 Pan improved the exponent from 2:81 to 2:79 [Pan78, Pan80]. One year later, Bini, Capovani, Lotti and Romani improved the exponent to 2:78 by constructing fast \approximative" algorithms for matrix multiplication and making these algorithms exact via the method of interpolation [BCRL79, Bin80]. Cast in the language of tensors, the result of Bini et al. corresponds to what we now call a \border rank" upper bound. The idea of studying approximative complexity or border complexity for algebraic problems has nowadays become an important theme in algebraic complexity theory. Sch¨onhagethen obtained the exponent 2:55 by constructing a fast algorithm for computing many \disjoint" small matrix multiplications and transforming this into an algorithm for one large matrix multiplication [Sch81]. The upper bound was improved shortly after by works of Pan [Pan81], Romani [Rom82], and Coppersmith and Winograd [CW82], resulting in the exponent 2:50. Then in 1987 Strassen published the laser method with which he obtained the expo- nent 2.48 [Str87]. The laser method was used in the same year by Coppersmith and Winograd to obtain the exponent 2.38 [CW87]. To do this they invented a method for constructing certain large combinatorial structures. This method, or actually the extended version that Strassen published in [Str91], we now call the Coppersmith{Winograd method. All further improvements on upper bounding 3 4 Chapter 1. Introduction the exponent essentially follow the framework of Coppersmith and Winograd, and the improvements do not affect the first two digits after the decimal point [CW90, Sto10, Wil12, LG14]. Define ! to be the smallest possible exponent of n in the cost of any matrix multiplication algorithm. (The precise definition will be given in Section 1.1.) We call ! the exponent of matrix multiplication. To summarise the above historical account on upper bounds: ! < 2:38. On the other hand, the only lower bound we currently have is the trivial lower bound 2 ≤ !. The history of upper bounds on the matrix multiplication exponent !, which began with Strassen's algorithm and ended with the Strassen laser method and Coppersmith{Winograd method, is well-known and well-documented, see e.g. [BCS97, Section 15.13]. However, there is remarkable work of Strassen on a theory of lower bounds for ! and similar types of exponents, and this work has received almost no attention in the literature. This theory of lower bounds is the theory of asymptotic spectra of tensors and is the topic of a series of papers by Strassen [Str86, Str87, Str88, Str91, Str05]. In the foregoing, the word tensor has popped up twice|namely, when we mentioned border rank and just now when we mentioned asymptotic spectra of tensors|but we have not discussed at all why tensors should be relevant for understanding the complexity of matrix multiplication. First, we give a mini course on tensors. A k-tensor t = (ti1;:::;ik )i1;:::;ik is a k-dimensional array of numbers from some field, say the complex numbers C. Thus, a 2-tensor is simply a matrix. A k-tensor is called simple if there exist k vectors v1; : : : ; vk such that the entries of t are given by the products ti1;:::;ik = (v1)i1 ··· (vk)ik for all indices ij.