The border support rank of two-by-two matrix multiplication is seven

Markus Blaeser,Matthias Christandl,Jeroen Zuiddam

doi:10.4086/cjtcs.2018.005

Abstract

We show that the border support rank of the tensor corresponding to two-by-two matrix multiplication is seven over the complex numbers. We do this by constructing two polynomials that vanish on all complex tensors with format four-by-four-by-four and border rank at most six, but that do not vanish simultaneously on any tensor with the same support as the two-by-two matrix multiplication tensor. This extends the work of Hauenstein, Ikenmeyer, and Landsberg. We also give two proofs that the support rank of the two-by-two matrix multiplication tensor is seven over any field: one proof using a result of De Groote saying that the decomposition of this tensor is unique up to sandwiching, and another proof via the substitution method. These results answer a question asked by Cohn and Umans. Studying the border support rank of the matrix multiplication tensor is relevant for the design of matrix multiplication algorithms, because upper bounds on the border support rank of the matrix multiplication tensor lead to upper bounds on the computational complexity of matrix multiplication, via a construction of Cohn and Umans. Moreover, support rank has applications in quantum communication complexity.

Highlights

Multiplication of two n × n matrices over a field F is an F-bilinear map Fn×n × Fn×n → Fn×n called the matrix multiplication map
(Technically, this is the structure tensor of the trilinear map that computes the trace of a product of three matrices.) Let V1, V2, and V3 be vector spaces
The tensor rank of a tensor t ∈ V1 ⊗V2 ⊗V3 is the smallest number r such that t can be written as a sum of r simple tensors v1 ⊗ v2 ⊗ v3 ∈ V1 ⊗V2 ⊗V3

Summary

Matthias Christandl*

Received June 14, 2017; Revised September 25, 2017, and in final form August 21, 2018; Published August 25, Abstract: We show that the border support rank of the tensor corresponding to two-bytwo matrix multiplication is seven over the complex numbers. We do this by constructing two polynomials that vanish on all complex tensors with format four-by-four-by-four and border rank at most six, but that do not vanish simultaneously on any tensor with the same support as the two-by-two matrix multiplication tensor. Key words and phrases: matrix multiplication, border support rank, algebraic complexity theory

Introduction

Support rank

Border support rank

Lower bounds by polynomials

The matrix multiplication tensor