“Under reasonable assumptions, we rigorously establish asymptotic speedups for our algorithms, relative to digital methods, that scale linearly in matrix dimension.”
The pitch is solving linear systems, inverses and determinants by letting coupled harmonic oscillators settle into equilibrium and sampling them. The paper dismisses quantum computing as too far off and offers thermodynamics as the near-term fix. The speedup rests on “reasonable assumptions” and on thermodynamic hardware that barely exists. It is a faster algorithm for a computer nobody can buy.