Adiabatically reducing quantum fluctuations guides the system to the ground state via quantum tunneling
Quantum Annealing Hamiltonian
The system evolves under:
H(t) = −Γ(t)Σᵢσᵢˣ − J Σ⟨ij⟩σᵢᶻσⱼᶻ
Starting with large Γ (quantum fluctuations dominate), then slowly decreasing to zero while increasing J. The ground state of H_x (all spins aligned in x) evolves adiabatically into the ground state of H_z (the optimization problem).
Adiabatic theorem: if Γ decreases slowly enough (T ≫ 1/Δ²), the system stays in the instantaneous ground state. The minimum gap Δ determines the required annealing time.
Quantum vs. Classical
Quantum tunneling allows the system to pass through energy barriers, while simulated annealing must thermally climb over them. This gives quantum annealing an advantage for rugged landscapes with many local minima separated by tall narrow barriers.
For the Ising spin glass H = −J Σ Jᵢⱼσᵢσⱼ, finding the ground state is NP-hard in general. Quantum annealing is implemented in physical hardware (D-Wave) using superconducting flux qubits.
P(ground state) ~ exp(−T_min/T)