A randomly walking quantum particle evolving by Schr\"odinger's equation searches for a unique marked vertex on the ``simplex of complete graphs'' in time $\mathrm{\ensuremath{\Theta}}({N}^{3/4})$. We give a weighted version of this graph that preserves vertex transitivity, and we show that the time to search on it can be reduced to nearly $\mathrm{\ensuremath{\Theta}}(\sqrt{N})$. To prove this, we introduce two extensions to degenerate perturbation theory: an adjustment that distinguishes the weights of the edges and a method to determine how precisely the jumping rate of the quantum walk must be chosen.
A randomly walking quantum particle evolving by Schr"odinger's equation searches for a unique marked vertex on the ``simplex of complete graphs'' in time \ensuremathΘ(N3/4). We give a weighted version of this graph that preserves vertex transitivity, and we show that the time to search on it can be reduced to nearly \ensuremathΘ(N). To prove this, we introduce two extensions to degenerate perturbation theory: an adjustment that distinguishes the weights of the edges and a method to determine how precisely the jumping rate of the quantum walk must be chosen.
A quantum particle evolving by Schr\"odinger's equation in discrete space constitutes a continuous-time quantum walk on a graph of vertices and edges. When a vertex is marked by an oracle, the quantum walk effects a quantum search algorithm. Previous investigations of this quantum search algorithm on graphs with cliques have shown that the edges between the cliques can be weighted to enhance the movement of probability between the cliques to reach the marked vertex. In this paper, we explore the most restrictive form of this by analyzing search on a weighted barbell graph that consists of two cliques of the same size joined by a single weighted edge or bridge. This graph is generally irregular, so quantum walks governed by the graph Laplacian or by the adjacency matrix can differ. We show that the Laplacian quantum walk's behavior does not change, no matter the weight of the bridge, and so the single bridge is too restrictive to affect the walk. Similarly, the adjacency quantum walk's behavior is unchanged for most weights, but when the weight equals the size of a clique, the probability does collect at the clique containing the marked vertex, and utilizing a two-stage algorithm with different weights for each stage, the success probability is boosted from 0.5 to 0.996, independent of the size of the barbell graph.
The simplex of complete graphs, also known as the first-order truncated simplex lattice, is a network of M+1 identical complete graphs, each with M vertices, such that each clique contains an edge or bridge to every other clique. It contains N=M(M+1) vertices, and previous asymptotic results using a continuous-time quantum walk to search this graph for a single marked vertex have either numerically demonstrated an optimal runtime of O(N), or analytically proved a deterministic success probability of 1, but not both, even when the bridges are weighted. In this paper, we give the first analytical proof of optimal quantum search on this graph, proving that it occurs when the weight of the bridges equals M. In addition, we numerically show that the optimal runtime is achieved more broadly whenever the weight is at least M. Furthermore, the algorithm is also deterministic when the weight scales between M and M, and this is the first example of quantum search on the simplex of complete graphs that is both asymptotically optimal and deterministic. In addition, for weights where the algorithm is nondeterministic, we give a way to find the marked vertex by inspecting neighboring vertices. Finally, while it is known that connectivity is not a reliable indicator of fast quantum search when comparing different graph families, we show that it is also unreliable within the graph family of weighted simplex of complete graphs.
Continuous-time quantum walks are natural tools for spatial search, where one searches for a marked vertex in a graph. Sometimes the structure of the graph causes the walker to get trapped, such that the probability of finding the marked vertex is limited. We give an example with two linked cliques, proving that the captive probability can be liberated by increasing the weights of the links. This allows the search to succeed with probability 1 without increasing the energy scaling of the algorithm. Further increasing the weights, however, slows the runtime, so the optimal search requires weights that are neither too weak nor too strong.
The coined quantum walk is a discretization of the Dirac equation of relativistic quantum mechanics, and it is the basis of many quantum algorithms. We investigate how it searches the complete bipartite graph of N vertices for one of k marked vertices with different initial states. We prove an intriguing dependence on the number of marked and unmarked vertices in each partite set. For example, when the graph is irregular and the initial state is the typical uniform superposition over the vertices, then the success probability can vary greatly from one time step to the next, even alternating between 0 and 1, so the precise time at which measurement occurs is crucial. When the initial state is a uniform superposition over the edges, however, the success probability evolves smoothly. As another example, if the complete bipartite graph is regular, then the two initial states are equivalent. Then if two marked vertices are in the same partite set, the success probability reaches 1/2, but if they are in different partite sets, it instead reaches 1. This differs from the complete graph, which is the quantum walk formulation of Grover's algorithm, where the success probability with two marked vertices is 8/9. This reveals a contrast to the continuous-time quantum walk, whose evolution is governed by Schr\"odinger's equation, which asymptotically searches the regular complete bipartite graph with any arrangement of marked vertices in the same manner as the complete graph.
We show how a quantum walk can be used to find a marked edge or a marked complete subgraph of a complete graph. We employ a version of a quantum walk, the scattering walk, which lends itself to experimental implementation. The edges are marked by adding elements to them that impart a specific phase shift to the particle as it enters or leaves the edge. If the complete graph has N vertices and the subgraph has K vertices, the particle becomes localized on the subgraph in O(N/K) steps. This leads to a quantum search that is quadratically faster than a corresponding classical search. We show how to implement the quantum walk using a quantum circuit and a quantum oracle, which allows us to specify the resources needed for a quantitative comparison of the efficiency of classical and quantum searches---the number of oracle calls.
Abstract We define a discrete-time, coined quantum walk on weighted graphs that is inspired by Szegedy’s quantum walk. Using this, we prove that many lackadaisical quantum walks, where each vertex has l integer self-loops, can be generalized to a quantum walk where each vertex has a single self-loop of real-valued weight l . We apply this real-valued lackadaisical quantum walk to two problems. First, we analyze it on the line or one-dimensional lattice, showing that it is exactly equivalent to a continuous deformation of the three-state Grover walk with faster ballistic dispersion. Second, we generalize Grover’s algorithm, or search on the complete graph, to have a weighted self-loop at each vertex, yielding an improved success probability when l < 3 + 2 2 ≈ 5.828 .