Ballistic spreading vs classical diffusion: interference determines quantum transport
t=0
Quantum walk (Aharonov et al 1993): Quantum version of random walk. Coin state (qubit) tensored with position. Hadamard coin: H|↑⟩=(|↑⟩+|↓⟩)/√2, H|↓⟩=(|↑⟩−|↓⟩)/√2. Ballistic spreading: σ(t) ~ t (vs classical σ ~ √t). Peak probability at ±t/√2 (not center). Quadratic speedup for quantum search algorithms. General coin: U(θ) = [[cos θ, sin θ],[sin θ, −cos θ]]. θ=45° = Hadamard. θ=0°: stays put. θ=90°: oscillates. Decoherence → classical: Adding noise destroys interference → recovers diffusive spreading. Boundary between quantum/classical transport. 2D applications: Grover's search, quantum transport in graphene, Anderson localization in disordered quantum walks.