Quantum Annealing Energy Landscape

Quantum tunneling through energy barriers vs thermal hopping over them. The transverse field Ising model: quantum fluctuations explore the landscape simultaneously via superposition.

Transverse field Γ (quantum): 1.0

Temperature T (thermal): 0.5

Barrier height V₀: 2.0

Barrier width W: 1.0

2D Energy Landscape — Transverse Field Ising Model

Tunneling Probability vs Barrier Width

0.42

Tunnel prob T(E)

0.08

Thermal hop P_SA

5.2×

QA advantage

Physics

H = -Γ·Σᵢσᵢˣ - J·Σᵢⱼσᵢᶻσⱼᶻ (Transverse Field Ising)

Tunneling: T ≈ exp(-2∫√(2m[V(x)-E]) dx/ℏ) [WKB approximation]

Thermal hopping: P = exp(-ΔV/kT) [Boltzmann]

D-Wave systems use ~5000 qubits. Quantum annealing can tunnel through barriers that are wide but thin — classical annealing must hop over them. QA advantage is strongest for structured problems with many thin barriers.