A quantum particle on a 1D line uses a "coin" unitary (Hadamard or general SU(2)) to spread its wavefunction. Unlike random walks, quantum walks spread ballistically with σ ~ t.
Step: 0 | σ (QW): — | σ (CRW): —
Coin operator: C(θ,φ) = [[cos θ, sin θ · e^{iφ}],[sin θ, -cos θ · e^{iφ}]]. Shift: |↑,x⟩→|↑,x+1⟩, |↓,x⟩→|↓,x-1⟩. Probability P(x,t) = |ψ↑|² + |ψ↓|². Classical RW shown in red.