In this paper we are going to discuss on the Quantum Computation approach and then we present a Quantum Algorithm for finding the estimation of the shortest path length between two vertices in a connected weighted graph with a polynomial time complexity O(n) whereas this problem based on the classic algorithm can be solved with time complexity about O(n2 -n).
In this paper we are going to discuss on the Quantum Computation approach and then we present a Quantum Algorithm for finding the estimation of the shortest path length between two vertices in a connected weighted graph with a polynomial time complexity O(n) whereas this problem based on the classic algorithm can be solved with time complexity about O(n2 -n).
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).
Quantum algorithms run on quantum computers are qualitatively different from those that run on classical computers. Quantum computing algorithms can be used for several problems in graph theory. Most of the classical algorithms involve searching over some space for finding the shortest-paths problem between two points in a graph and a minimal weight spanning tree. We modified classical Dijkstra's algorithm and implement quantum search instead of classical search, of which it will lead to more efficient algorithm. Also we proposed the structure for non-classical algorithms and design the various phases of the probabilistic quantum-classical algorithm for classical and quantum parts. Finally, we represent the result of implementing and simulating Dijkstra's algorithm as the probabilistic quantum-classical 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.