Quantum Walk on a Line — Ballistic Spreading

σ(t) ~ t (ballistic) vs σ(t) ~ √t (classical) — Hadamard coin, Anderson localization

Coin type

Coin angle θ 45°

Disorder W 0.0

Initial state

Steps per frame 5

Compare classical overlay

Quantum walk: coin ⊗ position space. Each step: apply coin C to internal state, then shift. Hadamard coin: H = (1/√2)[[1,1],[1,−1]]. Ballistic spreading: σ(t) ≈ t/√2 for symmetric initial state. Classical random walk: σ(t) = √t. Anderson localization: adding random phase disorder W collapses the wave packet — σ(t) saturates. Grover coin gives different interference pattern. Measurement probability: P(x,t) = |ψ_R(x,t)|² + |ψ_L(x,t)|².