[aifer4]

One way to think about these methods is that we are essentially implementing a Monte-Carlo algorithm physically, where on each "iteration" there is a matrix-vector multiplication. The physical system does this matrix-vector multiplication for us in constant time, so it does have an advantage over these digital methods. Not only that, but the "clock speed" of the physical system can be almost arbitrarily short, although this comes with an energy cost.

[fnordpiglet]

Yes that’s precisely what made me harken back to my solving of large stochastic PDE system questions :-)

The different though is instead of a matrix multiplication it’s a nonlinear optimization. The crucial part is the nonlinearity. But I assume given this technique is as you say Monte Carlo at its root, that shouldn’t specifically matter?

[aifer4]

You mentioned that such problems may be solved by solving a linear approximation of the non-linear problem (and I am in no way an expert in non-linear optimization). To the extent that the bottleneck in that approach is solving the resulting linear system, this method offers a speedup. We are also thinking about using similar thermodynamic methods to solve non-linear systems directly, but some of the nice properties of the harmonic oscillator are not present in that case, so it's currently not clear how much (if any) speedup is there.