Quantum Annealing — Adiabatic Gap & Landau-Zener Transitions

Adiabatic quantum computation evolves H(s) = (1−s)H_driver + s·H_problem. The key challenge: the spectral gap Δ(s) may close at a quantum phase transition, causing Landau-Zener diabatic excitations with probability P_LZ = exp(−πΔ²/2v̇) where v̇ is the sweep rate.

Energy levels E₀,E₁ vs annealing parameter s∈[0,1]. Minimum gap Δ_min = critical bottleneck.

Landau-Zener excitation probability P_LZ vs sweep rate. Slow annealing → adiabatic; fast → diabatic.

System size N: 6 spins Disorder Γ: 1.0 Annealing time T_ann: 10

Min gap Δ_min: —

Gap location s*: —

LZ prob P_LZ: —

Success prob: —

Required T_ann: —

Annealing schedules: linear, quadratic, optimal (Δ-weighted).

Ground state overlap with problem solution vs s.