Connectivity is a Poor Indicator of Fast Quantum Search

Abstract

A randomly walking quantum particle evolving by Schrödinger's equation searches on ddimensional cubic lattices in O( √ N ) time when d ≥ 5, and with progressively slower runtime as d decreases. This suggests that graph connectivity (including vertex, edge, algebraic, and normalized algebraic connectivities) is an indicator of fast quantum search, a belief supported by fast quantum search on complete graphs, strongly regular graphs, and hypercubes, all of which are highly connected. In this paper, we show this intuition to be false by giving two examples of graphs for which the opposite holds true: one with low connectivity but fast search, and one with high connectivity but slow search. The second example is a novel two-stage quantum walk algorithm in which the walking rate must be adjusted to yield high search probability. Introduction.-Despite ten years elapsing since the introduction of continuous-time quantum walk algorithms that search on graphs [1], there is still no comprehensive theory as to which graphs support fast quantum search. Nevertheless, much work has been done to further our understanding. For example, we recently showed that global symmetry is unnecessary for fast quantum search [2].Regarding specific graphs, a randomly walking quantum particle evolving by Schrödinger's equation searches on the complete graph, strongly regular graphs, and the hypercube in optimal Θ( √ N ) time, the first of which is precisely the continous-time analogue of Grover's algorithm [1][2][3][4]. Examples of these graphs are shown in Fig. 1. Additionally, such a particle can search on ddimensional cubic lattices in Θ( √ N ) total time when d ≥ 5, and with progressively slower runtimes as d decreases [1,5,6], as shown in Table I.One might suspect that fast search occurs when graphs are highly connected, as suggested by [1]. In this paper, however, we show this intuition to be false by giving two examples of graphs for which the opposite holds true: one with low connectivity but fast search, and one with high connectivity but slow search; they are shown in Figs. 2 and 3, respectively. To do this, we first introduce four different ways to measure graph connectivity. Then we detail how a randomly walking quantum particle searches on a graph. Finally, we determine the runtimes of our two examples.Measures of Connectivity.-The two most common ways to measure connectivity are vertex connectivity and edge connectivity, which are how many vertices or edges must be removed to make a graph disconnected. For example, Fig. 2 has vertex and edge connectivities of 1 because removing the yellow or green vertex disconnects the graph, and so does removing the edge between them. Note that vertex connectivity is upper bounded by the

Quantum spatial search on graphs subject to dynamical noise

“…Increasing the order does not improve the algorithm, on the contrary it provides worse performance, since the target degree remains the same while the possible connections with the central node increase, opening more "wrong ways" for the probability current going toward the target node. While connectivity seems to be irrelevant for noiseless quantum walks [12,13], our work points out that higher connectivity of the target node plays an important role in the presence of noise.…”

“…For instance, the algorithm has been investigated on complete bipartite graphs [5], on balanced trees [6], on Erdös-Rényi graphs [7,8], on the simplex of the complete graph [9] and on graphs with fractal dimensions [10,11]. Moreover, it has been shown that high connectivity and global symmetry of the graph are not necessary for fast quantum search [12,13]. The first result of this paper is a proof of the optimality of quantum spatial search on the star graph, both when the target is the central node and when it is one of the external ones.…”

Long-lived quantum speedup based on plasmonic hot spot systems

“…However, although the curves with 2 w and 3 w in figure 7(d) appear to have a long-lived quantum speedup (in the order of picosecond), the excitation can only be limited within the first two or three molecules (populations in figure 6(b)). Since the realization of quantum search algorithm needs to search in a large space [34][35][36], such confinement of excitation in some of molecules with 2 w and 3 w make our system do not suitable for realizing quantum search algorithm. Considering the discussion above, in the case with the radii of NP R=12 nm, the separation distance d=2 nm is an optimal solution for the ideal quantum walk with long-lived quantum speedup (compare figures 5 and 7).…”

“…Although the study on photonic waveguides has reported that the duration of running time reaches several picoseconds [30], such coupled waveguides are unable to realize three or higherdimensional continuous-time quantum walk for one of the spatial dimensions is used as propagation space (time). In comparison, due to the flexible construction within high-dimensional space [31][32][33], such plasmonic nanostructures can be designed to realize three or higher-dimensional continuous-time quantum walk, which can exhibit the quantum advantage over its correspondingly classical walk in executing the search algorithms [34][35][36].…”

Perfect state transfer by means of discrete-time quantum walk search algorithms on highly symmetric graphs

“…Moreover, including the coin degree of freedom the continuous-time quantum walk search is optimal for lattices with d > 2 [10]. Later it was found that high symmetry or connectivity of the graph is in fact not required for the optimal runtime of the continuoustime quantum walk search algorithm [11][12][13]. In fact, Chakraborty et al [14] have shown that continuous-time quantum walk search algorithm is optimal for almost all graphs.…”