Stationary States in Quantum Walk Search

TL;DR

This paper characterizes stationary states in quantum walk search algorithms, revealing how multiple marked vertices can hinder search efficiency and providing methods to identify and avoid such states.

Contribution

It offers a complete characterization of stationary states for quantum walk search operators on general graphs, including an optimization procedure and existence theorems.

Findings

Stationary states can be characterized by decomposing amplitudes into uniform and flip states.

The existence of stationary states depends on the bipartite nature of marked vertices.

Using a different oracle can prevent stationary states from hindering the search.

Abstract

When classically searching a database, having additional correct answers makes the search easier. For a discrete-time quantum walk searching a graph for a marked vertex, however, additional marked vertices can make the search harder by causing the system to approximately begin in a stationary state, so the system fails to evolve. In this paper, we completely characterize the stationary states, or 1-eigenvectors, of the quantum walk search operator for general graphs and configurations of marked vertices by decomposing their amplitudes into uniform and flip states. This infinitely expands the number of known stationary states and gives an optimization procedure to find the stationary state closest to the initial uniform state of the walk. We further prove theorems on the existence of stationary states, with them conditionally existing if the marked vertices form a bipartite connected component and always existing if non-bipartite. These results utilize the standard oracle in Grover's algorithm, but we show that a different type of oracle prevents stationary states from interfering with the search algorithm.