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You can make a natural correspondence between a quantum state vector and a classical probability vector, and between a quantum unitary operator and a classical stochastic matrix. There is also a correspondence between the quantum annealing algorithm and the classical simulated annealing algorithm. I am wondering whether it is possible to write down simulated annealing in the language of probability vectors and stochastic matrices, and then see what additional power is obtained by changing to the quantum counterparts.

More generally, I would like to bridge the language gaps between probabilistic algorithms and quantum algorithms, and I am wondering whether recasting probabilistic algorithms in terms of probability vectors and stochastic matrices has been tried before.

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Yes. This has been done by Morita and Nishimori in their 2008 publication, "Mathematical Foundations of Quantum Annealing."

https://arxiv.org/abs/0806.1859

In Section 5 they derive the convergence conditions from path integral Monte Carlo and Green function Monte Carlo methods. To quote;

In Sec. 5 we have derived the convergence condition of QA implemented by Quantum Monte Carlo simulations of path-integral and Green function methods. These approaches bear important practical significance because only stochastic methods allow us to treat practical large-size problems on the classical computer. A highly non-trivial result in this section is that the convergence condition for the stochastic methods is essentially the same power-law decrease of the transverse-field term as in the Schrödinger dynamics of Sec. 2. This is surprising since the Monte Carlo (stochastic) dynamics is completely different from the Schrödinger dynamics. Something deep may lie behind this coincidence and it should be an interesting target of future studies.

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