Stationary states in quantum walk 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.

Comment on Nahimovs et al. `On the probability of finding marked connected components using quantum walks'

“…Let G = (V, E) be an undirected graph and M be a connected set of marked vertices. In [1] the authors showed Theorem 1. If M is bipartite, then we can assign amplitudes to neutralise the shortages at each marked vertex if and only if the sums of the shortages on both partite sets are equal.…”

“…Note that authors have considered only stationary states with the amplitudes u → v for arbitrary unmarked u and arbitrary v being all equal. Since according to Theorem 2 from [1] only these amplitudes has impact on overlap between stationary state and initial state, a can be upper-bounded by ā of the form…”

Equivalence of Szegedy’s and coined quantum walks

“…Search using SKW's scheme has been explored on a large number of graphs, including the hypercube [1], two-and higher-dimensional grids [25,35,36], Sierpinski gaskets [37], complete graphs with and without self-loops [38], and the simplex of complete graphs with a fully marked clique [26]. Other explorations include the impact of internal-state measurements [39] and stationary states [40]. Now we prove that Szegedy's quantum walk with absorbing vertices is equivalent to SKW's coined quantum walk search scheme.…”

“…It first appeared in [25], where this search scheme on the complete graph with a self-loop at each vertex is exactly equivalent to Grover's algorithm (apart from a factor of 2). Other investigations of this quantum walk search operator SCQ include the complete graph with an arbitrary number of self-loops per vertex [38], search with potential barriers [32], improving the success probability by measuring the coin state [39], and the effect of stationary states [40]. All of these results directly map to Santos's walk W q1 .…”

Exceptional quantum walk search on the cycle

“…For example, consider the discrete-time coined quantum walk that was first proposed by Meyer in the context of quantum cellular automata [6,7] and later recast as a quantum walk by Aharonov et al [8]. Its ability to search the 2D grid has been explored for several arrangements of marked vertices [9,10,11,12,13,14], and one problematic configuration is when the marked vertices lie along a diagonal, such as in Fig. 1b.…”