Published by the Department of Computer Science, The University of Chicago.

Saarland University

Saarbrücken

Germany

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 `

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.

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Submitted June 14, 2017, revised September 25, 2017, and August 21, 2018,
published August 28, 2018.

© 2018 Markus Bläser, Matthias Christandl, and Jeroen Zuiddam

Licensed under a Creative Commons Attribution License

Licensed under a Creative Commons Attribution License

Volume 2018, Article 4

Volume 2018 Published articles