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We describe two algorithms for multiplying n x n matrices using time andenergy n^2 polylog(n) under basic models of classical physics. The firstalgorithm is for multiplying integer-valued matrices, and the second, quitedifferent algorithm, is for Boolean matrix multiplication. We hope this workinspires a deeper consideration of physically plausible/realizable models ofcomputing that might allow for algorithms which improve upon the runtimes andenergy usages suggested by the parallel RAM model in which each operationrequires one unit of time and one unit of energy.

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