Matrix Multiplication in Quadratic Time and Energy? Towards a Fine-Grained Energy-Centric Church-Turing Thesis

Author Gregory Valiant

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Gregory Valiant
  • Department of Computer Science, Stanford University, CA, USA


I want to thank Vijaykrishna Gurunathan for discussions on this topic. Vijaykrishna wished to not be included as an author.

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Gregory Valiant. Matrix Multiplication in Quadratic Time and Energy? Towards a Fine-Grained Energy-Centric Church-Turing Thesis. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 96:1-96:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We describe two algorithms for multiplying n × n matrices using time and energy Õ(n²) under basic models of classical physics. The first algorithm is for multiplying integer-valued matrices, and the second, quite different algorithm, is for Boolean matrix multiplication. We hope this work inspires a deeper consideration of physically plausible/realizable models of computing that might allow for algorithms which improve upon the runtimes and energy usages suggested by the parallel RAM model in which each operation requires one unit of time and one unit of energy.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Design and analysis of algorithms
  • Physics based computing
  • matrix multiplication
  • low-energy computing


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