Full Characterization of Oscillatory Localization of Quantum Walks

TL;DR

This paper fully characterizes the conditions under which quantum walks exhibit oscillatory localization, where the walker alternates between states, using simple graph examples and state classifications.

Contribution

It introduces the concept of oscillatory localization in quantum walks and provides a complete characterization of the states that exhibit this behavior.

Findings

Oscillatory localization occurs in specific quantum states.

Only flip states and uniform states exhibit exact oscillatory localization.

Projection onto these states bounds the extent of localization.

Abstract

Discrete-time quantum walks are well-known for exhibiting localization, a quantum phenomenon where the walker remains at its initial location with high probability. In companion with a joint Letter, we introduce oscillatory localization, where the walker alternates between two states. The walk is given by the flip-flop shift, which is easily defined on non-lattice graphs, and the Grover coin. Extremely simple examples of the localization exist, such as a walker jumping back and forth between two vertices of the complete graph. We show that only two kinds of states, called flip states and uniform states, exhibit exact oscillatory localization. So the projection of an arbitrary state onto these gives a lower bound on the extent of oscillatory localization. Thus we completely characterize the states that oscillate under the quantum walk.