Quantum Walk vs Classical Random Walk

A quantum particle in superposition spreads quadratically faster than its classical counterpart — the key advantage of quantum computing.

Quantum walk (Hadamard coin) Classical random walk (averaged)
Steps: 0 Quantum std: 0 Classical std: 0 Speedup: -
The quantum walk uses a Hadamard coin: |↑⟩ → (|↑⟩+|↓⟩)/√2, |↓⟩ → (|↑⟩-|↓⟩)/√2. Interference between left- and right-moving components causes the characteristic double-peaked distribution. Spread σ∝t vs classical σ∝√t — quadratic speedup.