I'm studing how Quantum Metropolis Algorithm (QMA) works and I think that I've understood it. Generally, the basic Metropolis step for a gave hamiltonian at the inverse temperature $\beta$ is: 1) it starts with a initial state, 2) it changes it with a local random "kick operator", 3) it uses quantum phase estimation for evaluate the energy gap beetwen the initial state and the new state, 4) it applies a controlled rotation (based on the energy gap) to rotate an ancilla qubit prepared in a computational base state, 5) and finally it mesures this state to accept/reject the proposal new state.
First of all, I ask confirms wether this vague explanation is correct or not. Then I'll ask you the main question: in many articles on QMA, authors add to the general explanation of the Algorithm also a part where they use Quantum Simulated Annealing (QSA). I understand that this procedure is useful to switch from an initial state with high temperature to a state with lower, but I can't understand why this process is necessary: this "temperature correction" has not already done by the QMA previus described?
Maybe QSA is only an explanation of why QMA actually works...
Thank you in advance!