Quantum Walk: Coined Delocalization

Discrete-time quantum walk (DTQW) is the quantum analog of a random walk. A "coin" qubit is flipped by a unitary C (Hadamard, Grover, etc.), then the walker steps left/right based on coin state: |x,↑⟩→|x+1,↑⟩, |x,↓⟩→|x−1,↓⟩. Quantum interference produces ballistic spreading σ∝T, quadratically faster than classical diffusion σ∝√T. The probability distribution shows two peaks near ±T/√2 with quantum interference fringes between them. Adding disorder (decoherence) drives the walk toward classical diffusive behavior — Anderson localization at strong disorder.