Jeroen Zuiddam

Nondeterministic Quantum Communication Complexity: the Cyclic Equality Game and Iterated Matrix Multiplication

We study nondeterministic multiparty quantum communication with a quantum generalization of broadcasts. We show that, with number-in-hand classical inputs, the communication complexity of a Boolean function in this communication model equals the logarithm of the support rank of the corresponding tensor, whereas the approximation complexity in this model equals the logarithm of the border support rank. This characterisation allows us to prove a log-rank conjecture posed by Villagra et al. for nondeterministic multiparty quantum communication with message-passing. The support rank characterization of the communication model connects quantum communication complexity intimately to the theory of asymptotic entanglement transformation and algebraic complexity theory. In this context, we introduce the graphwise equality problem. For a cycle graph, the complexity of this communication problem is closely related to the complexity of the computational problem of multiplying matrices, or more precisely, it equals the logarithm of the asymptotic support rank of the iterated matrix multiplication tensor. We employ Strassens laser method to show that asymptotically there exist nontrivial protocols for every odd-player cyclic equality problem. We exhibit an efficient protocol for the 5-player problem for small inputs, and we show how Young flattenings yield nontrivial complexity lower bounds.

A note on the gap between rank and border rank

We study the tensor rank of the tensor corresponding to the algebra of n-variate complex polynomials modulo the dth power of each variable. As a result we find a sequence of tensors with a large gap between rank and border rank, and thus a counterexample to a conjecture of Rhodes. At the same time we obtain a new lower bound on the tensor rank of tensor powers of the generalised W-state tensor. In addition, we exactly determine the tensor rank of the tensor cube of the three-party W-state tensor, thus answering a question of Chen et al.

Asymptotic tensor rank of graph tensors: beyond matrix multiplication

We present an upper bound on the exponent of the asymptotic behaviour of the tensor rank of a family of tensors defined by the complete graph on k vertices. For

Tensor rank is not multiplicative under the tensor product

{k \geq 4}

On Algebraic Branching Programs of Small Width

k≥4, we show that the exponent per edge is at most 0.77, outperforming the best known upper bound on the exponent per edge for matrix multiplication (k = 3), which is approximately 0.79. We raise the question whether for some k the exponent per edge can be below 2/3, i.e. can outperform matrix multiplication even if the matrix multiplication exponent equals 2. In order to obtain our results, we generalize to higher-order tensors a result by Strassen on the asymptotic subrank of tight tensors and a result by Coppersmith and Winograd on the asymptotic rank of matrix multiplication. Our results have applications in entanglement theory and communication complexity.

Clean Quantum and Classical Communication Protocols

The asymptotic restriction problem for tensors s and t is to find the smallest β ≥ 0 such that the nth tensor power of t can be obtained from the (β n+o(n))th tensor power of s by applying linear maps to the tensor legs — this is called restriction — when n goes to infinity. Applications include computing the arithmetic complexity of matrix multiplication in algebraic complexity theory, deciding the feasibility of an asymptotic transformation between pure quantum states via stochastic local operations and classical communication in quantum information theory, bounding the query complexity of certain properties in algebraic property testing, and bounding the size of combinatorial structures like tri-colored sum-free sets in additive combinatorics. Naturally, the asymptotic restriction problem asks for obstructions (think of lower bounds in computational complexity) and constructions (think of fast matrix multiplication algorithms). Strassen showed that for obstructions it is sufficient to consider maps from k-tensors to nonnegative reals, that are monotone under restriction, normalised on diagonal tensors, additive under direct sum and multiplicative under tensor product, named spectral points (SFCS 1986 and J. Reine Angew. Math. 1988). Strassen introduced the support functionals, which are spectral points for oblique tensors, a strict subfamily of all tensors (J. Reine Angew. Math. 1991). On the construction side, an important work is the Coppersmith-Winograd method for tight tensors and tight sets. We present the first nontrivial spectral points for the family of all complex tensors, named quantum functionals. Finding such universal spectral points has been an open problem for thirty years. We use techniques from quantum information theory, invariant theory and moment polytopes. We present comparisons among the support functionals and our quantum functionals, and compute generic values. We relate the functionals to instability from geometric invariant theory, in the spirit of Blasiak et al. (Discrete Anal. 2017). We prove that the quantum functionals are asymptotic upper bounds on slice-rank and multi-slice rank, extending a result of Tao and Sawin. Furthermore, we make progress on the construction side of the combinatorial version of the asymptotic restriction problem by extending the Coppersmith–Winograd method via combinatorial degeneration. The regular method constructs large free diagonals in powers of any tight set. Our extended version works for any set that has a combinatorial degeneration to a tight set. This generalizes a result of Kleinberg, Sawin and Speyer. As an application we reprove in hindsight recent results on tri-colored sum-free sets by reducing this problem to a result of Strassen on reduced polynomial multiplication. Proofs are in the full version of this paper, available at https://arxiv.org/abs/1709.07851.

Geometric complexity theory

The tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an l-tensor. The tensor product of s and t is a (k+l)-tensor. Tensor rank is sub-multiplicative under the tensor product. We revisit the connection between restrictions and degenerations. A result of our study is that tensor rank is not in general multiplicative under the tensor product. This answers a question of Draisma and Saptharishi. Specifically, if a tensor t has border rank strictly smaller than its rank, then the tensor rank of t is not multiplicative under taking a sufficiently hight tensor product power. The “tensor Kronecker product” from algebraic complexity theory is related to our tensor product but different, namely it multiplies two k-tensors to get a k-tensor. Nonmultiplicativity of the tensor Kronecker product has been known since the work of Strassen. It remains an open question whether border rank and asymptotic rank are multiplicative under the tensor product. Interestingly, lower bounds on border rank obtained from generalized flattenings (including Young flattenings) multiply under the tensor product.

The asymptotic spectrum of graphs and the Shannon capacity.

We introduce a method for transforming low-order tensors into higher-order tensors and apply it to tensors defined by graphs and hypergraphs. The transformation proceeds according to a surgery-like procedure that splits vertices, creates and absorbs virtual edges and inserts new vertices and edges. We show that tensor surgery is capable of preserving the low rank structure of an initial tensor decomposition and thus allows to prove nontrivial upper bounds on tensor rank, border rank and asymptotic rank of the final tensors. We illustrate our method with a number of examples. Tensor surgery on the triangle graph, which corresponds to the matrix multiplication tensor, leads to nontrivial rank upper bounds for all odd cycle graphs, which correspond to the tensors of iterated matrix multiplication. In the asymptotic setting we obtain upper bounds in terms of the matrix multiplication exponent ω and the rectangular matrix multiplication parameter α. These bounds are optimal if ω equals two. We also give examples that illustrate that tensor surgery on general graphs might involve the absorption of virtual hyperedges and we provide an example of tensor surgery on a hypergraph. Besides its relevance in algebraic complexity theory, our work has applications in quantum information theory and communication complexity.