Asymptotic tensor rank of graph tensors: beyond matrix multiplication

TL;DR

This paper establishes new upper bounds on the asymptotic tensor rank for graph tensors, surpassing matrix multiplication bounds for certain cases, with implications for entanglement and communication complexity.

Contribution

It introduces improved upper bounds on the tensor rank exponent for complete graph tensors with at least four vertices, extending previous results and generalizing key tensor rank theorems.

Findings

Exponent per edge for k≥4 is at most 0.77

Outperforms the matrix multiplication exponent per edge (≈0.79)

Raises open questions about surpassing the 2/3 exponent threshold

Abstract

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 $k\geq4$, 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 generalise 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.