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Quantum Walks and Structured Searches on Free Groups and Networks

Michael C. Ratner

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

Quantum walks have been utilized by many quantum algorithms which provide improved performance over their classical counterparts. Quantum search algorithms, the quantum analogues of spatial search algorithms, have been studied on a wide variety of structures. We study quantum walks and searches on the Cayley graphs of finitely-generated free groups. Return properties are analyzed via Green’s functions, and quantum searches are examined. Additionally, the stopping times and success rates of quantum searches on random networks are experimentally estimated.

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arXiv:quant-ph/0503078•Journal of Physics A Mathematical and General•Dec 21, 2005•27 citations
Quantum walks have been utilized by many quantum algorithms which provide improved performance over their classical counterparts. Quantum search algorithms, the quantum analogues of spatial search algorithms, have been studied on a wide variety of structures. We study quantum walks and searches on the Cayley graphs of finitely-generated free groups. Return properties are analyzed via Green’s functions, and quantum searches are examined. Additionally, the stopping times and success rates of quantum searches on random networks are experimentally estimated.
Quantum walk is the quantum analogue of the classical random walk. Its application in search has demonstrated the capability to solve certain types of problems more efficiently than classical methods. The quantum walk search on the dihedral Cayley graph with 2N vertices is examined under several generating sets S\mathcal {S}S. With the minimal generating set, the results show that the layout is an exceptional configuration. By adding a self-loop edge with some weight w > 0, which coincides with the lackadaisical quantum walk framework, and extra edges for each vertex, a nearly constant success probability of around 0.80 can be obtained using the proposed weight w′=1Nw^{\prime }=\frac{1}{N}w′=N1​. The fitted number of steps is given by τ′ = 1.221N − 0.391. A comparison with the cycle graph is made, demonstrating improvement in the expected number of steps under the same vertex set size.
This paper explores the application of quantum walk algorithms to graph traversal and search problems. Classical algorithms often struggle with the computational complexity of traversing and searching large graphs, leading to significant time and resource limitations. Quantum walks, utilizing superposition and interference, offer a potential pathway to achieve exponential speedups. This work details the theoretical framework for applying quantum walks to graph problems, specifically focusing on traversal and search tasks. We present a generalized approach for mapping graph structures onto quantum walk systems and discuss the key mathematical formulations involved. The core of our investigation revolves around the construction of quantum walk operators that mimic the movement and interaction rules inherent in graph traversal. We analyze the evolution of the quantum walk state and derive expressions for the probability of reaching a specific node in the graph. Furthermore, we examine the impact of graph topology on the performance of the quantum walk, highlighting the potential for tailored algorithms to exploit specific graph structures. The results demonstrate the theoretical feasibility of quantum walk algorithms for significantly accelerating graph traversal and search operations, contributing to advancements in areas such as network analysis, database management, and artificial intelligence.
Lackadaisical quantum walks are coined quantum walk models in which each vertex is associated with a weighted self-loop, allowing the walker to remain at its current position. They have been shown to effectively enhance spatial search on various graphs under different oracle models, with the cyclic graph being the most fundamental instance. In this paper, lackadaisical quantum walk search is studied on Cayley graphs of dicyclic groups, a non-abelian extension of cyclic groups, under three different oracle models. A preliminary analysis of the search performance is first conducted on dicyclic Cayley graphs with respect to the minimal generating set of the underlying dicyclic group. By expanding the generating sets to enable symmetric exploration and applying an appropriate oracle, the quantum walk search achieves a near-one success probability. Under our proposed configurations, the expected runtime to search on dicyclic Cayley graphs outperforms both quantum walk search on the cycle graph and classical unstructured search.
Quantum random walks on graphs have been shown to display many interesting properties, including exponentially fast hitting times when compared with their classical counterparts. However, it is still unclear how to use these novel properties to gain an algorithmic speedup over classical algorithms. In this paper, we present a quantum search algorithm based on the quantum random-walk architecture that provides such a speedup. It will be shown that this algorithm performs an oracle search on a database of N items with O(N)O(\sqrt{N})O(N​) calls to the oracle, yielding a speedup similar to other quantum search algorithms. It appears that the quantum random-walk formulation has considerable flexibility, presenting interesting opportunities for development of other, possibly novel quantum algorithms.
Spatial search and walk processes. Spatial search by quantum walk is database search with the additional constraint that one is required to move through the search space that obeys some locality structure. For example, the data items may be stored at the vertices of a two-dimensional grid. The requirement of moves along the edges of the grid captures the cost of accessing different items starting from some fixed position in the database. One of possible ways of carrying out spatial search is by performing a random walk on the search space or its quantum analog, a quantum walk. The complexity of spatial search by quantum walk is strongly tied to the quantum hitting time [19] of the walk. Let S , with jS j D n, be a finite set of states. Assume that a subset M S of states are marked. We are given a procedure C that, on input x 2 S and an associated data structure d.x/, checks whether the state x is marked. The goal is either to find a marked state when promised that M ¤ ; (search version) or to determine whether M is nonempty (decision version). The algorithm progresses in stages. In the setup stage, we access some state of S (usually a random state). In the walk stage we move from state to state, performing a spatial walk as described below. The moves are called updates. In addition, in the walk stage we perform checks to see if the current state is marked at steps selected by the algorithm. In the classical setting, the transition probabilities of the spatial walk are described by a stochastic matrix P D .px;y/x;y2S . This makes the walk a Markov chain. In every move the algorithm must perform a random transition according to P . The possible x ! y moves, i.e., those with px;y ¤ 0, form the edges of a (directed) graph G, and we say that the Markov chain P has locality structure G. We define the search problem in the classical setting, which carries over to the quantum case with little modification:
We address the problem of the construction of quantum walks on Cayley graphs. Our main motivation is the relationship between quantum algorithms and quantum walks. In particular, we discuss the choice of the dimension of the local Hilbert space and consider various classes of graphs on which the structure of quantum walks may differ. We completely characterize quantum walks on free groups and present partial results on more general cases. Some examples are given including a family of quantum walks on the hypercube involving a Clifford algebra.