Complex to Rational Fast Matrix Multiplication

TL;DR

The paper addresses the practical challenge that many fast matrix-multiplication schemes rely on complex coefficients, which hinder efficiency. It develops a De-Groote-action framework that yields a necessary-and-sufficient condition for when a complex-valued scheme admits an equivalent real- or rational-coefficient form, and it provides an explicit algorithm to decide and construct such equivalents. The authors recover prior results (e.g., Dumas–Pernet–Sedoglavic) and establish new non-existence results, such as Smirnov’s scheme over $\mathbb{Q}(\sqrt{161})$ having no equivalent rational form and Kaporin’s complex scheme having no equivalent real form, while also extending the framework to integer-coefficient questions. Through applications to various known tensors, the work clarifies which complex schemes can be realized over simpler rings and highlights the practical impact for implementing fast matrix multiplication with favorable arithmetic. Overall, this framework guides the selection and transformation of fast algorithms for real-world, hardware-friendly computation.

Abstract

Fast matrix multiplication algorithms are asymptotically faster than the classical cubic-time algorithm, but they are often slower in practice. One important obstacle is the use of complex coefficients, which increases arithmetic overhead and limits practical efficiency. This paper focuses on transforming complex-coefficient matrix multiplication schemes into equivalent real- or rational-coefficient ones. We present a systematic method that, given a complex-coefficient scheme, either constructs a family of equivalent rational algorithms or proves that no equivalent rational scheme exists. Our approach relies only on basic linear-algebraic properties of similarity transformations of complex matrices. This method recovers the previously known ad hoc results of Dumas, Pernet, and Sedoglavic (2025) and extends them to more general settings, including algorithms involving rational coefficients and square roots, with $i=\sqrt{-1}$ as a special case. Using this framework, we show that no rational scheme is equivalent to Smirnov's $\langle4,4,9,104\rangle$ $\mathbb{Q}[\sqrt{161}]$ algorithm (2022) and that no real scheme is equivalent to the $\langle4,4,4,48\rangle$ complex algorithm of Kaporin (2024). More generally, our approach can also be used to prove the non-existence of integer-coefficient schemes.

Theorems & Definitions (17)

• Proposition 1
• Proposition 2
• Lemma 1
• proof
• Theorem 1
• Theorem 2
• Lemma 2
• proof
• Lemma 3
• proof