Ballistic quantum spreading vs diffusive classical spreading in 1D
The discrete quantum walk uses a Hadamard coin: (|↑⟩+|↓⟩)/√2 tensored with position shift operators S± = Σ|x±1⟩⟨x|. After t steps the standard deviation σ_Q ~ t (ballistic), compared to σ_C ~ √t (classical). The coin angle θ determines the asymmetry; θ=45° (Hadamard) gives the characteristic two-peak structure. Quantum walks underlie Grover search and quantum transport in photosynthesis.