Standard Grover quantum walks concentrate amplitude on a goal node in a struc- tured graph, making them useful for maze solving. However, in graphs with cycles, interference from non-optimal paths reduces goal probability. This paper proposes an enhanced Grover walk framework tailored to unicyclic structures. The approach mod- ifies the projection operator and navigation vector to mitigate destructive interference. Empirical evaluation using classical and quantum simulations demonstrates improved convergence, reduced sink absorption, and up to 20% higher goal probabilities. The technique generalizes to other low-cycle graphs like ladders, offering a robust tool for quantum-assisted search. The technique is further extended to ladder graphs. Keywords: quantum search, grover walks, probability-based search
Standard Grover quantum walks concentrate amplitude on a goal node in a struc- tured graph, making them useful for maze solving. However, in graphs with cycles, interference from non-optimal paths reduces goal probability. This paper proposes an enhanced Grover walk framework tailored to unicyclic structures. The approach mod- ifies the projection operator and navigation vector to mitigate destructive interference. Empirical evaluation using classical and quantum simulations demonstrates improved convergence, reduced sink absorption, and up to 20% higher goal probabilities. The technique generalizes to other low-cycle graphs like ladders, offering a robust tool for quantum-assisted search. The technique is further extended to ladder graphs. Keywords: quantum search, grover walks, probability-based search
Abstract We provide a mathematical framework for identifying the shortest path in a maze using a Grover walk, which becomes non-unitary by introducing absorbing holes. In this study, we define the maze as a network with vertices connected by unweighted edges. Our analysis of the stationary state of the truncated Grover walk on finite graphs, where we strategically place absorbing holes and self-loops on specific vertices, demonstrates that this approach can effectively solve mazes. By setting arbitrary start and goal vertices in the underlying graph, we obtain the following long-time results: (i) in tree structures, the probability amplitude is concentrated exclusively along the shortest path between start and goal; (ii) in ladder-like structures with additional paths, the probability amplitude is maximized near the shortest path.
Quantum walks offer promising advantages for search algorithms over graphs. Among these, Grover’s quantum search provides a quadratic speedup with a time complexity of for unstructured search problems. Nevertheless, Grover search performs poorly on cyclic graphs due to the dynamics of the Grover walk. This study explores the behavior of Grover’s quantum walk on cyclic graphs , analyzing the probability distribution of finding the marked vertex. To analyze this behavior, we extend each vertex of the cycle by attaching semi-infinite-length paths (tails). We develop a direct analytical approach to obtain the transition matrix , whose elements are independent of . For small cycles , we observe success probabilities exceeding 0.5. In contrast, with large cycles, the effectiveness drops rapidly. We observe that the success probability approaches zero as the size of the graph increases , indicating the limitations of the quantum search in cyclic structures.
This paper explores the potential of quantum-inspired algorithms for graph search problems. Traditional random walk approaches often suffer from inefficiencies, particularly in complex graphs. We propose a novel framework that leverages concepts from quantum walks – specifically interference and tunneling – within a classical computational environment. Our algorithms dynamically adjust the probability distribution of node visits based on graph connectivity and the history of previous steps, mimicking the probabilistic nature of quantum mechanics. We demonstrate that this approach can, in certain scenarios, outperform standard random walks, offering a more adaptable and potentially faster search strategy. The core of the method lies in approximating the quantum effects through carefully designed classical simulations, without requiring actual quantum hardware. We present a mathematical formulation of the core algorithm and discuss its key parameters, focusing on the balance between classical simulation fidelity and computational cost. The results suggest a promising direction for graph search optimization, drawing inspiration from the principles of quantum mechanics.
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.
Algorithms and data structures are important in information technology (particularly in 6G terahertz wireless communications), logistics, transportation, and route planning in medicine. One of the fundamental algorithms that find application in many practical problems is the shortest path algorithm. The shortest path algorithm is used to find the optimal route between two vertices while minimizing the length or number of intermediate nodes. This problem is one of the 21 NP-hard problems formulated by Richard Karp in 1972. It still poses a serious challenge to science, since there is no polynomial algorithm that can provide an exact solution to this problem. The article considers approaches to solving the problem of finding the shortest path on a graph, and discusses their advantages and disadvantages. Various algorithms, such as the Bellman-Ford algorithm, Dijkstra, and the A* algorithm, are considered; their efficiency and applicability in various situations are analyzed. Issues of optimization of shortest path algorithms are discussed, such as the use of heuristics, graph preprocessing, and other methods that can improve the performance and accuracy of algorithms. The article proposes an optimization of Dijkstra’s algorithm with quantum acceleration. The stability of the quantum algorithm and the magnitude of distortions in a circuit with a different number of qubits are estimated. It is shown that with an increase in the intensity of distortions, the probability of detecting the desired element decreases, and the time spent on its search increases. When using Grover’s algorithm to optimize a path in a quantum network, these interferences cause an increase in the time it takes to find the optimal path and an increase in the probability of choosing an incorrect route. It is shown that an increase in the number of qubits and, accordingly, the number of iterations makes the algorithm more stable. It is demonstrated that the stability of Grover’s algorithm for circuits with a large number of qubits has a similar nature. The work of quantum circuits on 9, 10 and 11 qubits, the difference in the probabilities of correct operation differs by less than 10–3. Thus, from the obtained dependencies, it is possible to predict the nature of such an influence on circuits with an even larger number of qubits, the stability of which cannot be studied due to the insufficient computing power of classical computers. The most efficient way to modify Dijkstra’s algorithm is to replace the priority queue in the classical version of the algorithm with some quantum data structure that finds the minimum element for the next iteration of the algorithm. Despite the fact that quantum search does not provide a potential gain in asymptotics, another quantum modification of Dijkstra’s algorithm can be used. Namely, instead of maintaining and searching for a minimum in the priority queue, use the quantum algorithm immediately to search for a vertex that needs to be added to the shortest path tree, which will additionally acceleration the graph traversal by Dijkstra’s algorithm.
In Mobile Ad-hoc Networks (MANETs), routing protocols directly affect various indices of network Quality of Service (QoS), so they play an important role in network performance. To address the drawbacks associated with traditional routing protocols in MANETs, such as poor anti-fading performance and slow convergence rate, for basic Dynamic Source Routing (DSR), propose a new routing model based on improved Grover’s searching algorithm. In this paper, a hybrid approach that combines quantum walks with Grover search is proposed so that the search is accelerated and optimum is found with fewer numbers of iterations. By taking advantage of quantum tunneling effect, local barriers are overcome and better threshold values can be found, especially at the early stage of search process. The new algorithm based on the formalism of continuous-time quantum walk is demonstrated. With this new routing model, each node maintains a node vector function, and all the nodes can obtain a node probability vector using improved Grover’s algorithm, and then select an optimal routing according to node probability. Simulation results show that compared with DSR, this new routing protocol can effectively extend the network lifetime, as well as reduce the network delay and the number of routing hops. The results between the new algorithm and the state-of-the-art Grover search method are also compared.