1D discrete-time quantum walk — interference and ballistic spreading
The 1D quantum walk uses a qubit "coin" (Hadamard by default) applied each step, with a conditional shift operator:
S shifts |↑⟩ right, |↓⟩ left. Unlike classical random walks (σ ~ √t), quantum walks spread ballistically:
This quadratic speedup is key to quantum search algorithms (Grover: O(√N) vs O(N)). The asymmetric double-peaked distribution is due to quantum interference. Symmetric initial states yield symmetric distributions.