### Volume 2018

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

Markus Bläser
Saarland University
Saarbrücken
Germany
mblaeser AT cs DOT uni-saarland DOT de

Matthias Christandl
University of Copenhagen
Copenhagen, Denmark
christandl AT math DOT ku DOT dk

and

Jeroen Zuiddam
CWI Amsterdam
and
University of Amsterdam
Amsterdam, Netherlands
j DOT zuidamm AT cwi DOT nl

August 25, 2018

#### 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 may be interpreted as a quantum communication complexity measure.

• The article: PDF (256 KB)
• Source material: ZIP (115 KB) Note: contains also Python code used in calculations.