A conventional computer computes by shuttling electric charges along wires. But can you compute with heat? A network of pipes carrying hot and cold water can perform simple logic operations if the pipes are connected in the right order. Temperature differences act like valves in a processor, except instead of voltage, you have heat differences.

Such thermal computers have already been proposed, but they come at a price: each computation requires dissipating some heat, otherwise the signal drowns in random temperature fluctuations. The authors of the new work derived rigorous formulas linking the accuracy of the answer, the information transfer rate, and the inevitable energy losses. You cannot completely eliminate errors: even with very large heat expenditures, a minimum noise level remains, set by the size of the thermal reservoirs.

When several thermal elements are linked in a chain to perform a complex task, the energy cost grows not just proportionally to the number of elements. The authors showed that for a network of length L, the losses grow as L times the logarithm of L, and with strong noise amplification, up to L cubed. Large thermal circuits become impractically power-hungry, and one must find ways to cool or isolate each stage.

The work bridges the physics of thermal processes and information theory. It gives engineers clear benchmarks: how much energy will have to be spent for a thermal computer to compute with the required accuracy, and where lies the boundary beyond which it is pointless to go.