In this paper we evaluate our presented Quantum Approach for finding the Estimation of the Length of the Shortest Path in a Connected Weighted Graph which is achieved with a polynomial time complexity about O(n) and as a result of evaluation we show that the Probability of Success of our presented Quantum Approach is increased if the Standard Deviation of the Length of all possible paths between Source and Target vertices is increased also we show that if the Standard Deviation is low nevertheless our Quantum Computation Approach is satisfactory and even the standard deviation is exactly zero the probability of Success of our Quantum Approach is not only zero but also is %100. Also we present a Classic Parallel Approach for finding the Estimation of the Length of the shortest path and then as a comparison between our Quantum Computation and Classic Parallel Approaches we show that even by applying the Classic Parallel Approach the problem can be solved with time complexity about O(log(2n -1)).
In this paper we evaluate our presented Quantum Approach for finding the Estimation of the Length of the Shortest Path in a Connected Weighted Graph which is achieved with a polynomial time complexity about O(n) and as a result of evaluation we show that the Probability of Success of our presented Quantum Approach is increased if the Standard Deviation of the Length of all possible paths between Source and Target vertices is increased also we show that if the Standard Deviation is low nevertheless our Quantum Computation Approach is satisfactory and even the standard deviation is exactly zero the probability of Success of our Quantum Approach is not only zero but also is %100. Also we present a Classic Parallel Approach for finding the Estimation of the Length of the shortest path and then as a comparison between our Quantum Computation and Classic Parallel Approaches we show that even by applying the Classic Parallel Approach the problem can be solved with time complexity about O(log(2n -1)).
Given an undirected, weighted graph, with n vertices and m edges, and two special vertices s and t, the problem is to find the shortest path between them. We give two bounded-error quantum algorithms with improved runtime in the adjacency list model that solve the problem on special classes of graphs defined via pathfinding probabilities of classical random walks and the electrical network framework. Firstly, we give a simple quantum algorithm based on sampling edges from a graph via the quantum flow state and running a classical algorithm on the sampled edges. It runs in O~(l2m) expected time and uses O(logn) space on graphs where the shortest s-t path is also a minimum resistance s-t subgraph. Our main algorithm can be thought of as a divide and conquer version of this approach and works on a special class of graphs where classical loop-erased random walk has a probability q>0.537 of finding the shortest s-t path. In such cases the quantum algorithm outputs the shortest s-t path with high probability in O(ℓm) expected time and O(logn) space, where l is the length (or total weight, in case of weighted graphs) of the shortest s-t path. This algorithm can be parallelised to O~(lm) circuit depth when using O(llogn) space. With the latter we partially resolve with an affirmative answer the open problem of whether a path between two vertices can be found in the number of steps required to detect it.
The problem of finding the shortest path between two points in a graph is a fundamental problem in computer science and has many applications in fields such as transportation, logistics, and networking. Dijkstra's algorithm is a classical algorithm commonly used to solve this problem, but its time complexity can be prohibitively high for large graphs. Quantum computing, on the other hand, offers a promising approach to solving this problem with significantly improved time complexity. In this research paper, we explore the potential of quantum computing for solving the shortest path problem, including an overview of Dijkstra's algorithm and its limitations, the quantum computing approach using quantum phase estimation, and a comparison of the time complexities of classical and quantum algorithms. We also discuss the challenges that need to be addressed for quantum computing to realize its potential in solving practical shortest-path problems.
In this work, we consider a well-known “Single Source Shortest Path Search” problems for weighted directed acyclic graphs (DAGs). We suggest a quantum algorithm with time complexity O(nmlogn) and O(1/n) error probability, where n is a number of Vertexes, m is the number of edges. We use quantum dynamic programming approach (Khadiev, 2018) and Dürr and Høyer minimum search algorithm to speed up our search. Our algorithm is better than C. Dürr and coauthors’ quantum algorithm for an undirected graph. The time complexity of C. Dürr’s algorithm is O(nm(logn)2) . The best known deterministic algorithm for the problem is based on a dynamic programming approach and its time complexity is O (n + m). At the same time, if we use algorithms for general graphs, then we do not get better results. The time complexity of best implementations of Dijkstra’s algorithm with Fibonacci heap is O (m + n log n).
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.
We provide a rigorous theoretical analysis of quantum algorithms for shortest path problems in weighted graphs, establishing complexity-theoretic foundations for quantum acceleration in this fundamental computational domain. Building upon recent quantum adaptations of classical graph algorithms, we develop a unified framework that characterizes when and why quantum approaches offer computational advantages for single-source and all-pairs shortest path computations. Our analysis encompasses both positive-weight and negative-weight scenarios, deriving conditional speedup theorems that depend on graph structure, weight distribution, and oracle access models. We prove that quantum amplitude amplification and quantum walks can provide polynomial to exponential speedups under specific structural assumptions, while establishing hardness results that identify problem instances where quantum advantage is provably absent. The theoretical framework introduces novel complexity measures that capture graph-theoretic properties relevant to quantum algorithm performance, including quantum diameter, entanglement capacity, and interference efficiency. We extend our analysis to dynamic graph scenarios where edge weights evolve over time, establishing bounds on quantum advantage preservation under graph modifications. This work contributes fundamental insights into quantum algorithmic graph theory, clarifying the boundaries of quantum computational advantage and providing theoretical guidance for algorithm design in network optimization, routing, and graph analytics applications.