Discrete Quantum Walk on Integer Line

Ballistic spreading σ~t vs classical diffusive σ~√t; interference from coin superposition
80
45°
0.50
Discrete quantum walk: walker on ℤ with internal coin state |↑⟩,|↓⟩. Coin operator: C(θ) = [[cos θ, i sin θ],[i sin θ, cos θ]] (unitary). Shift: S|↑,x⟩=|↑,x+1⟩, S|↓,x⟩=|↓,x−1⟩. Quantum interference produces ballistic spreading σ~t (vs classical σ~√t). Hadamard coin (θ=45°) gives asymmetric distribution from unequal initial conditions.