Exceptional quantum walk search on the cycle

Abstract

Quantum walks are standard tools for searching graphs for marked vertices, and they often yield quadratic speedups over a classical random walk's hitting time. In some exceptional cases, however, the system only evolves by sign flips, staying in a uniform probability distribution for all time. We prove that the one-dimensional periodic lattice or cycle with any arrangement of marked vertices is such an exceptional configuration. Using this discovery, we construct a search problem where the quantum walk's random sampling yields an arbitrary speedup in query complexity over the classical random walk's hitting time. In this context, however, the mixing time to prepare the initial uniform state is a more suitable comparison than the hitting time, and then the speedup is roughly quadratic.

Equivalence of Szegedy’s and coined quantum walks

“…Note that |φ x is the equal superposition of edges incident to x ∈ X, and |ψ y is the equal superposition of edges incident to y ∈ Y . As proved in [27] and in congruence with the "inversion about the mean" of Grover's algorithm [23], the reflection R 1 goes through each vertex in X and reflects the amplitude of its incident edges about their average amplitude, and R 2 similarly does this for the vertices in Y . For example, for the bipartite double cover in Fig.…”

“…At unmarked vertices, R 1 and R 2 act identically to R 1 and R 2 , respectively, by inverting the amplitudes of the edges around their average at each vertex. At marked vertices, however, R 1 and R 2 act differently [27], flipping the signs of the amplitudes of all incident edges.…”

“…2b. As shown in [27], the dashed line connecting vertex 2 in X and Y can be ignored since it has zero amplitude throughout the evolution, so the effective graph is Fig. 2c.…”

“…At unmarked vertices, R 1 and R 2 act identically to R 1 and R 2 , respectively, by inverting the amplitudes of the edges around their average at each vertex. At marked vertices, however, R 1 and R 2 act differently [27], flipping the signs of the amplitudes of all incident edges. This seminal search scheme has been investigated for a variety of graphs, including the one-dimensional periodic lattice or cycle [27,28] and the complete graph [29].…”

“…At marked vertices, however, R 1 and R 2 act differently [27], flipping the signs of the amplitudes of all incident edges. This seminal search scheme has been investigated for a variety of graphs, including the one-dimensional periodic lattice or cycle [27,28] and the complete graph [29]. Szegedy also investigated his search scheme for general symmetric and vertex-transitive graphs with a unique marked vertex [14].…”