Both searches are random walkers proposing small jumps along a 1D energy curve V(x) built from several valleys separated by barriers. Neither can see the whole landscape — they only feel the local slope of each proposed move.
Classical (simulated annealing): a single marble accepts an uphill proposal with Metropolis probability exp(−ΔV/T), where ΔV is the full height of the barrier along the path. T cools every step, so late in the run only tiny uphill moves are still possible. Crossing depends only on how tall the barrier is — width is irrelevant.
Quantum (quantum annealing): an ensemble of walkers (the probability cloud) gets a second, independent acceptance channel: a WKB-style tunneling probability exp(−A·width·√height)·Γ(t). This depends on both barrier width and height, but width enters linearly while height only enters as a square root — so a barrier that is tall but thin can still be tunneled through even after Γ(t), the annealing "transverse field", has mostly decayed.
P_classical(cross) = exp(-height_gap / T)
P_quantum(cross) = 1-(1-P_classical)(1-Γ·exp(-A·width·sqrt(height_gap)))
- Tall & thin barrier: classical needs a lucky high-energy jump that gets rarer as T cools — it usually freezes in a mediocre valley. Quantum's tunneling term barely notices the height and slips through.
- Wide & short barrier: tunneling probability collapses (width dominates the exponent), so quantum loses its edge — but classical does fine here too, since a short barrier costs little thermal energy to climb.
- This is exactly the caveat real quantum annealers face: tunneling helps specifically against tall-narrow barriers, not against every hard landscape.