Physical systems can compute. If you build a landscape of hills and valleys so that the lowest point corresponds to the answer, the system will roll there by itself, like a ball. The problem is that the bottom is often flat, and the ball crawls around for a long time before settling.
The authors propose nudging the ball to the right place before it even starts. A classical computer quickly estimates which motions in the system are the slowest and prepares the initial state so that these slow motions barely occur. The physical system then covers the rest of the path on its own, but without unnecessary wandering.
The paper considers simple cases — inverted parabolas and systems with friction. For these, they could calculate exactly how much the waiting time shrinks. For example, in matrix inversion, the gain depends on the spread of eigenvalues: the more they differ, the more noticeable the speedup.
This hybrid approach — digital calculation plus physical relaxation — makes thermodynamic computing more practical. Instead of waiting for the system to find its own path, we hand it a map that avoids the swamps.
The Mpemba effect, which inspired the authors, is still debated: in some experiments hot water freezes faster, in others it doesn't. But in physics it has found new life as a way to speed up computing.
In Stanislaw Lem's story 'The Fourteenth Voyage', the hero encounters a machine that runs on Brownian motion — the random jostling of molecules. Optimizing the start could make such a machine much more efficient.