Discrete-time quantum walk — interference vs classical diffusion
Quantum walk |ψ|² Classical random walk
Discrete quantum walk: A quantum particle on a line with an internal spin (coin) degree of freedom. At each step, a unitary coin operator C(θ) rotates the spin, then the particle shifts right (↑) or left (↓). The Hadamard coin (θ=45°) creates the famous bimodal distribution — quantum interference causes probability to pile up at the edges rather than the Gaussian center seen in classical diffusion. The spread scales as σ ~ t (ballistic) vs σ ~ √t classically.