The quantum search on the graph is a very important topic. In this work, we develop a theoretic method on searching of single vertex on the graph [Phys. Rev. Lett. 114 110503 (2015)], and systematically study the search of many vertices on one low-connectivity graph, the joined complete graph. Our results reveal that, with the optimal jumping rate obtained from the theoretical method, we can find such target vertices at the time O ( N ) , where N is the number of total vertices. Therefore, the search of many vertices on the joined complete graph possessing quantum advantage has been achieved.
The quantum search on the graph is a very important topic. In this work, we develop a theoretic method on searching of single vertex on the graph [Phys. Rev. Lett. 114 110503 (2015)], and systematically study the search of many vertices on one low-connectivity graph, the joined complete graph. Our results reveal that, with the optimal jumping rate obtained from the theoretical method, we can find such target vertices at the time O ( N ) , where N is the number of total vertices. Therefore, the search of many vertices on the joined complete graph possessing quantum advantage has been achieved.
n this paper, we consider the problem of searching a marked vertex in a graph based on quantum walks. A very large body of research exists on this issue, from both theoretical and experimental studies, yet the existing quantum walk-based search algorithms require knowing exactly how many marked vertices there are before starting the search process. In addition, even if the precise knowledge of the number of marked vertices is known, the success probability of these algorithms could shrink dramatically when the number of search steps is greater than the right one. For avoiding these defects, in this paper we present a new quantum walk-based search algorithm which need not know any prior information about the marked vertices, but still keeps a quadratic speedup over classical ones and ensures that the error is bounded by a tunable parameter ϵ. More specifically, for an N-vertex complete bipartite graph with marked vertices but without knowing the number of marked vertices, if the number of search steps h satisfies h≥ln(ϵ2)N, then the algorithm will output a marked vertex with probability at least $ 1-\epsilon$ for any given ϵ∈(0,1]. This feature leads to quantum search algorithms with stronger robustness and a wider scope of application.
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.
Quantum walks are a potent technique for building quantum algorithms. This paper examines the quantum walk search algorithm on complete multipartite graphs with multiple marked vertices, which has not been explored before. We employ the coined quantum walk model and achieve quadratic speedup with a constant probability of finding a marked vertex in two specific cases of complete multipartite graphs. Furthermore, we investigate the robust quantum walk of two cases and demonstrate that even with an unknown number of marked vertices, it is still possible to achieve a quadratic speedup compared to classical algorithms and the success probability oscillates within a small range close to 1. To establish this conclusion, we employ approximation and reduction methods to conduct theoretical analysis and proofs, and provide an intuitive demonstration through numerical simulations. This work addresses the overcooking problem in quantum walk search algorithms on some complete multipartite graphs and may offer insights for robust quantum search on broader graph structures. Finally, we design a circuit for the algorithm that future researchers can use to conduct experiments.
Quantum walk is a potent technique for building quantum algorithms. This paper examines the quantum walk search algorithm on complete multipartite graphs with multiple marked vertices, which has not been explored before. Two specific cases of complete multipartite graphs are probed in this paper, and in both cases, each set consists of an equal number of vertices. We employ the coined quantum walk model and achieve quadratic speedup with a constant probability of finding a marked vertex. Furthermore, we investigate the robust quantum walk of two cases and demonstrate that even with an unknown number of marked vertices, it is still possible to achieve a quadratic speedup compared to classical algorithms and the success probability oscillates within a small range close to 1. This work addresses the overcooking problem in quantum walk search algorithms on some complete multipartite graphs. We also provide the numerical simulation and circuit implementation of our quantum algorithm.
The quantum walk is a powerful tool to develop quantum algorithms, which usually are based on searching for a vertex in a graph with multiple marked vertices, with Ambainis's quantum algorithm for solving the element distinctness problem being the most shining example. In this work, we address the problem of calculating analytical expressions of the time complexity of finding a marked vertex using quantum-walk-based search algorithms with multiple marked vertices on arbitrary graphs, extending previous analytical methods based on Szegedy's quantum walk, which can be applied only to bipartite graphs. Two examples based on the coined quantum walk on two-dimensional lattices and hypercubes show the details of our method.
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.