  We describe two algorithms for multiplying n x n matrices using time and
energy n^2 polylog(n) 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.
