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

TL;DR

This paper proves that the border support rank of the 2x2 matrix multiplication tensor is seven over complex numbers, providing new bounds relevant for matrix multiplication algorithms and quantum communication complexity.

Contribution

It establishes the exact border support rank of the 2x2 matrix multiplication tensor as seven, extending previous work and providing two proofs over any field.

Findings

Border support rank of 2x2 matrix multiplication tensor is seven over complex numbers.

Constructed polynomials that distinguish tensors with support similar to the matrix multiplication tensor.

Provides two proofs of the rank over any field, using decomposition uniqueness and substitution method.

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.