Quantum Walk — Discrete Grover Coin
Ballistic spread vs classical diffusion; bimodal probability distribution
A discrete quantum walk on a line: a coin operator (2×2 unitary) acts on the internal spin, then a shift operator moves the walker. The Hadamard/Grover coin at θ=45° produces ballistic spreading σ~t vs classical σ~√t. The probability distribution is bimodal (peaks near ±t/√2). The coin state creates interference — |+⟩ initial state gives asymmetric distribution; complex initial state |i⟩ gives symmetric bimodal peaks.