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Lobachevskii Journal of Mathematics

On the Probability of Finding Marked Connected Components Using Quantum Walks

Kamil Ravilevich Khadiev, Nikolajs Nahimovs, Raqueline A. M. Santos

Published Sep 1, 2018• 5 citations
Publisher

Abstract

Finding a marked vertex in a graph can be a complicated task when using quantum walks. Recent results show that for two or more adjacent marked vertices search by quantum walk with Grover’s coin may have no speed-up over classical exhaustive search. In this paper, we analyze the probability of finding a marked vertex for a set of connected components of marked vertices. We prove two upper bounds on the probability of finding a marked vertex and sketch further research directions.

DOI
10.1134/s1995080218070144

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Finding a marked vertex in a graph can be a complicated task when using quantum walks. Recent results show that for two or more adjacent marked vertices search by quantum walk with Grover’s coin may have no speed-up over classical exhaustive search. In this paper, we analyze the probability of finding a marked vertex for a set of connected components of marked vertices. We prove two upper bounds on the probability of finding a marked vertex and sketch further research directions.
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.
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\epsilonϵ. More specifically, for an NNN-vertex complete bipartite graph with marked vertices but without knowing the number of marked vertices, if the number of search steps hhh satisfies h≥ln⁡(2ϵ)Nh \geq \ln(\frac{2}{\sqrt{\epsilon}})\sqrt{N}h≥ln(ϵ​2​)N​, then the algorithm will output a marked vertex with probability at least $ 1-\epsilon$ for any given ϵ∈(0,1]\epsilon\in (0,1]ϵ∈(0,1]. This feature leads to quantum search algorithms with stronger robustness and a wider scope of application.
There has been a very large body of research on searching a marked vertex on a graph based on quantum walks, and Grover's algorithm can be regarded as a quantum walk-based search algorithm on a special graph. However, the existing quantum walk-based search algorithms suffer severely from the souffl\'e problem, which mainly means that the success probability of finding a marked vertex could shrink dramatically, even to zero, when the number of search steps is greater than the right one, thus heavily reducing the robustness and practicability of the algorithm. Surprisingly, while the souffl\'e problem of Grover's algorithm has attracted enough attention, how to address this problem for general quantum walk-based search algorithms is missing in the literature. Here we initiate the study of overcoming the souffl\'e problem for quantum walk-based search algorithms by presenting a quantum walk-based search framework that achieves robustness without sacrificing the quantum speedup. In this framework, for any adjustable parameter $\ensuremath{\epsilon}$, the quantum algorithm can find a marked vertex on an NNN-vertex complete bipartite graph with probability at least $1\ensuremath{-}\ensuremath{\epsilon}$, whenever the number of search steps hhh satisfies $h\ensuremath{\ge}ln(\frac{2}{\sqrt{\ensuremath{\epsilon}}})\sqrt{N}+1$. Note that the algorithm need not know the exact number of marked vertices. Consequently, we obtain quantum search algorithms with stronger robustness and practicability.
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.
The lazy random walk, where the walker has some probability of staying put, is a useful tool in classical algorithms. We propose a quantum analogue, the lackadaisical quantum walk, where each vertex is given l self-loops, and we investigate its effects on Grover's algorithm when formulated as search for a marked vertex on the complete graph of N vertices. For the discrete-time quantum walk using the phase flip coin, adding a self-loop to each vertex boosts the success probability from 1/2 to 1. Additional self-loops, however, decrease the success probability. Using instead the Shenvi, Kempe, and Whaley (2003) coin, adding self-loops simply slows down the search. These coins also differ in that the first is faster than classical when l scales less than N , while the second requires that l scale less than N 2 . Finally, continuous-time quantum walks differ from both of these discrete-time examples—the self-loops make no difference at all. These behaviors generalize to multiple marked vertices.