Quantum vs Classical Random Walk
Two-component |L>/|R> amplitude vector on a 101-site lattice, Hadamard coin every step, position probability summed as |psiL|^2 + |psiR|^2. Side-by-side classical binomial histogram for comparison.
WHAT TO TRY
- Read the two space-time carpets P(x,t) side by side: the classical walk is a narrow diffusive cone (width grows with sqrt of the steps), the quantum walk a wide ballistic triangle whose interior lattice of interference fringes funnels probability into two horns at the light-cone edge.
- Scrub the steps slider and watch the spread readout: classical sigma climbs as sqrt(t) while quantum sigma climbs linearly in t, so the speed-up ratio keeps growing. That quadratic advantage is why quantum walks power fast search algorithms.
- Switch the coin state from symmetric to left- or right-biased: the same Hadamard rule now sends the ballistic distribution skewing to one side, a sensitivity to initial conditions the classical walk never shows.
- That quadratic speedup is why quantum walks underpin fast search algorithms, the wave nature of the walker doing the work.