Quantum annealing on a 2-qubit Ising problem. H(s) = (1-s)H_x + s·H_z. Watch the ground state evolve as s goes 0→1, and how the gap closes near avoided crossings. Compare with classical simulated annealing.
Adiabatic theorem (Born-Fock 1928): stay in ground state if T >> ℏ/Δ_min² where Δ_min is the minimum gap. H(s) interpolates from transverse field (easy ground state: all ↑) to problem Hamiltonian. Quantum speedup requires gap closing polynomially vs exponentially (NP-hard problems). D-Wave machines implement this. Adiabatic QC is equivalent to circuit QC (Aharonov et al. 2007).