Localization of Two-Dimensional Five-State Quantum Walks

Understand

We investigate a generalized Hadamard walk in two dimensions with five inner states.

• The particle governed by a five-state quantum walk (5QW) moves, in superposition, either leftward, rightward, upward, or downward according to the inner state.
• In addition to the four degrees of freedom, it is allowed to stay at the same position.
• We calculate rigorously the wave function of the particle starting from the origin in the plane for any initial state, and give the spatial distribution of probability of finding the particle.