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arXiv:2312.03768

Algoritmo de Contagem Quântico Aplicado ao Grafo Bipartido Completo

Gustavo Alves Bezerra

Published Dec 5, 2023• 0 citations
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Abstract

Studies on Quantum Computing have been developed since the 1980s, motivating researches on quantum algorithms better than any classical algorithm possible. An example of such algorithms is Grover's algorithm, capable of finding $k$ (marked) elements in an unordered database with $N$ elements using $O(\sqrt{N/k})$ steps. Grover's algorithm can be interpreted as a quantum walk in a complete graph (with loops) containing $N$ vertices from which $k$ are marked. This interpretation motivated search algorithms in other graphs -- complete bipartite graph, grid, and hypercube. Using Grover's algorithm's linear operator, the quantum counting algorithm estimates the value of $k$ with an error of $O(\sqrt{k})$ using $O(\sqrt{N})$ steps. This work tackles the problem of using the quantum counting algorithm for estimating the value $k$ of marked elements in other graphs; more specifically, the complete bipartite graph. It is concluded that for a particular case, running the proposed algorithm at most $t$ times wields an estimation of $k$ with an error of $O(\sqrt{k})$ using $O(t\sqrt{N})$ steps and success probability of at least $(1 - 2^{-t})8/π^2$.

DOI
10.48550/arxiv.2312.03768

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Studies on Quantum Computing have been developed since the 1980s, motivating researches on quantum algorithms better than any classical algorithm possible. An example of such algorithms is Grover's algorithm, capable of finding kkk (marked) elements in an unordered database with NNN elements using O(N/k)O(\sqrt{N/k})O(N/k​) steps. Grover's algorithm can be interpreted as a quantum walk in a complete graph (with loops) containing NNN vertices from which kkk are marked. This interpretation motivated search algorithms in other graphs -- complete bipartite graph, grid, and hypercube. Using Grover's algorithm's linear operator, the quantum counting algorithm estimates the value of kkk with an error of O(k)O(\sqrt{k})O(k​) using O(N)O(\sqrt{N})O(N​) steps. This work tackles the problem of using the quantum counting algorithm for estimating the value kkk of marked elements in other graphs; more specifically, the complete bipartite graph. It is concluded that for a particular case, running the proposed algorithm at most ttt times wields an estimation of kkk with an error of O(k)O(\sqrt{k})O(k​) using O(tN)O(t\sqrt{N})O(tN​) steps and success probability of at least (1−2−t)8/π2(1 - 2^{-t})8/π^2(1−2−t)8/π2.
Quantum counting is a key quantum algorithm that aims to determine the number of marked elements in a database. This algorithm is based on the quantum phase estimation algorithm and uses the evolution operator of Grover’s algorithm because its nontrivial eigenvalues are dependent on the number of marked elements. Since Grover’s algorithm can be viewed as a quantum walk on a complete graph, a natural way to extend quantum counting is to use the evolution operator of quantum-walk-based search on noncomplete graphs instead of Grover’s operator. In this paper, we explore this extension by analyzing the coined quantum walk on the complete bipartite graph with an arbitrary number of marked vertices. We show that some eigenvalues of the evolution operator depend on the number of marked vertices and using this fact we show that the quantum phase estimation can be used to obtain the number of marked vertices. The time complexity for estimating the number of marked vertices in the bipartite graph with our algorithm equates closely with that of the original quantum counting 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.
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 NNN vertices for one of kkk 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 1/21/21/2, 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 8/98/98/9. 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.
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.
In a fundamental paper [Phys. Rev. Lett. 78, 325 (1997)] Grover showed how a quantum computer can find a single marked object in a database of size NNN by using only O(N)O(\sqrt{N})O(N​) queries of the oracle that identifies the object. His result was generalized to the case of finding one object in a subset of marked elements. We consider the following computational problem: A subset of marked elements is given whose number of elements is either MMM or KKK, our task is to determine which is the case. We show how to solve this problem with a high probability of success using iterations of Grover's basic step only, and no other algorithm. Let mmm be the required number of iterations; we prove that under certain restrictions on the sizes of MMM and KKK the estimation $m<2\sqrt{N}∕(\sqrt{K}\ensuremath{-}\sqrt{M})$ obtains. This bound reproduces previous results based on more elaborate algorithms, and is known to be optimal up to a constant factor. Our method involves simultaneous Diophantine approximations, so that Grover's algorithm is conceptualized as an orbit of an ergodic automorphism of the torus. We comment on situations where the algorithm may be slow, and note the similarity between these cases and the problem of small divisors in classical mechanics.
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.