Complexity: simulated annealing vs. quantum annealing How do I compare the performance of simulated annealing against the performance of quantum annealing algorithms?

In Convergence theorems for quantum annealing by Morita and Nishimori, it has been shown that QA performs better than SA. Let's me quote from the Discussion section:


  We have proved strong ergodicity of the inhomogeneous Markov chains
  associated with Quantum Annealing using both Path Integral Monte Carlo
  and Green's Function Monte Carlo methods, mainly with the application
  to the Transverse Field Ising Model in mind, which covers a wide range
  of combinatorial optimization problems. Our proof is quite general in
  the sense that it does not depend on the spatial dimensionality or the
  lattice structure of the system. The convergence of QA is guaranteed
  if the transverse field decreases as $\Gamma(t) \approx
 \frac{const}{t^c}$ asymptotically. This annealing schedule for the
  transverse field is faster than the temperature-annealing schedule,
  the log-inverse law, found by Geman and Geman for Simulated Annealing.


Does this result settle the question of the performances for Simulated vs. Quantum annealings? I have a feeling that we should put a little more context here.

I assume to settle the question we need to find a quantum annealing schedule which is faster than all simulated annealing schedules. The problem can be reduced to finding a quantum annealing schedule which is faster than the fastest simulated annealing schedule. Now, to me, the fastest simulated annealing schedule makes sense only when we know what optimization problem we are talking about. We cannot find the fastest simulated annealing schedule for all conceivable optimization problems. That just doesn't make sense to me.

So, is the performance comparison between SA and QA settled? Is yes, how? If no, how can we do it?
In Rough Large Deviation Estimates for Simulated Annealing: Application to Exponential Schedules a theoretical justification for the exponential cooling schedule was given. The results there basically state that for a fixed computation time N you can always choose an appropriate constant $ C= f(N)$  for an exponential cooling schedule $ \beta = \beta_0 \cdot \exp \left( n f(N) \right) $ with $ f(N) \sim N^{-1} \ln\left(N\right) $, such that simulated annealing will converge. So in light of these results I would say it is clear that simulated annealing can converge faster than quantum annealing, at least for the quantum annealing schedule result you cited. Although, it seems you have to tweak the cooling schedule to the computation time.

A nice overview of theoretically grounded acceleration of simulate annealing is given here Theoretically Grounded Acceleration Techniques for Simulated Annealing