The quantum walk Hamiltonian H = -γL + |w⟩⟨w| (oracle). Quantum amplitude spreads via the graph Laplacian L, with a marked vertex |w⟩ creating constructive interference.
Farhi-Gutmann (1998) continuous-time QW:
i d|ψ⟩/dt = H|ψ⟩, H = -γL - |w⟩⟨w|
On complete graph: optimal γ*=1/N, then P(w,t*)→1 at t*=π√N/2.
Quantum speedup: O(√N) vs classical O(N).
Amplitude evolves under Schrödinger equation. Interference between graph eigenstates selectively amplifies the marked vertex |w⟩, analogous to Grover's algorithm.
Vertex colors: probability amplitude |⟨v|ψ(t)⟩|².