Dynamic Quantum Walk
A quantum walk is the quantum-mechanical analog of a simple classical random walk. In the classical 1D model, a particle steps left or right with equal probability. A quantum walk, in contrast, evolves in superposition, and different paths can interfere.
For a classical random walk, the position distribution after n steps is derived from a binomial distribution and becomes approximately Gaussian for large n. A quantum walk typically yields a very different shape: interference can suppress some positions and amplify others, often leading to an overall faster spread than in the classical case.
The (discrete-time, coined) quantum walk is governed by a unitary step operator U acting on the combined Hilbert space of position and coin:
\[U = S \cdot (C \otimes I)\]where C is the coin operator (often chosen as the Hadamard gate) and S is the conditional shift operator.
This visualization shows how the measurement probabilities over position evolve with time, highlighting the contrast between classical diffusion and quantum interference.