Quantum Walk on 1D Lattice

Coin operator + shift. Quantum walk spreads as O(t) vs classical O(√t). Watch the interference pattern!

Steps t50

Speed3

Coin angle θ45°

Init state

Quantum

Classical

Classical RW: σ ∝ √t
Quantum walk: σ ∝ t
Quantum is quadratically faster!
Peaks at ±t/√2 (ballistic transport)

State: |ψ⟩ = Σ (αₓ|x,↑⟩+βₓ|x,↓⟩)
Coin: U(θ) = [[cosθ, sinθ],[sinθ, -cosθ]]
Shift: S|x,↑⟩ = |x+1,↑⟩
S|x,↓⟩ = |x-1,↓⟩
P(x,t) = |αₓ|²+|βₓ|²