Oscillatory localization of quantum walks analyzed by classical electric circuits

Abstract

We examine an unexplored quantum phenomenon we call oscillatory localization, where a discretetime quantum walk with Grover's diffusion coin jumps back and forth between two vertices. We then connect it to the power dissipation of a related electric network. Namely, we show that there are only two kinds of oscillating states, called uniform states and flip states, and that the projection of an arbitrary state onto a flip state is bounded by the power dissipation of an electric circuit. By applying this framework to states along a single edge of a graph, we show that low effective resistance implies oscillatory localization of the quantum walk. This reveals that oscillatory localization occurs on a large variety of regular graphs, including edge-transitive, expander, and high degree graphs. As a corollary, high edge-connectivity also implies localization of these states, since it is closely related to electric resistance.

Coined quantum walks on weighted graphs

“…If the graph is unweighted, so w vu = 1 for all u ∼ v, thenᾱ v is the average of the amplitudes pointing from v to its neighbors. Then the amplitudes are inverted about their mean (see Lemma 2 of [22]), and this is akin to the "inversion about the mean" of Grover's algorithm [16]. With weights,ᾱ v is no longer the mean, so the coin is no longer an inversion about the mean.…”

Stationary states in quantum walk search

“…In this section, we begin by introducing the concepts of uniform and flip states from [8], which form an orthogonal basis for directional states. Then we give conditions for a state to be stationary under the quantum walk search operator U (2).…”

“…Note that [8] defined uniform and flip states with regard to all vertices, whereas we only define them here with regards to individual vertices.…”

“…C is the coin flip that applies the Grover diffusion coin 2|s c s c | − I [7] on the directional space at each vertex, where |s c = dv i=1 |i / √ d v is the equal superposition over the directions. As shown in Lemma 2 of [8], the Grover coin inverts each amplitude of the directional state about the average amplitude, so it is the "inversion about the mean" of Grover's algorithm [2]. Finally, Q is an oracle query that flips the sign of the amplitude at marked vertices, which is the standard oracle in Grover's algorithm.…”

“…In doing so, we greatly expand the number of known stationary configurations to an infinite class. We do this by incorporating concepts from [8] regarding uniform and flip states, which form an orthogonal basis for directional states. Then we show that the type of stationary state described by Nahimovs, Rivosh, and Santos [10,11] is optimal, in the sense that it is the stationary states closest to the initial uniform state (1).…”