Classical algorithms have been used to search over some space for finding the shortest paths problem between two points in a network and a minimal weight spanning tree for routing. Any classical algorithm deterministic or probabilistic will clearly used O(N) steps since on the average it will measure a large fraction of N records. Quantum algorithm is the fastest possible algorithm that can do several operations simultaneously due to their wave like properties. This wave gives an O( N ) steps quantum algorithm for identifying that record, where was used classical Dijkstra’s algorithm for finding shortest path problem in the graph of network and implement quantum search. 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.
Classical algorithms have been used to search over some space for finding the shortest paths problem between two points in a network and a minimal weight spanning tree for routing. Any classical algorithm deterministic or probabilistic will clearly used O(N) steps since on the average it will measure a large fraction of N records. Quantum algorithm is the fastest possible algorithm that can do several operations simultaneously due to their wave like properties. This wave gives an O( N ) steps quantum algorithm for identifying that record, where was used classical Dijkstra’s algorithm for finding shortest path problem in the graph of network and implement quantum search. 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.
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.
Quantum computing algorithms are considered for several problems in graph theory. Classical algorithms involve searching over some space for finding the minimal spanning tree problem in a graph. We modify classical Primpsilas algorithm and replace 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 Primpsilas algorithm in the probabilistic quantum-classical algorithm.
Quantum algorithms for several problems in graph theory are considered. Classical algorithms for finding the lowest weight path between two points in a graph and for finding a minimal weight spanning tree involve searching over some space. Modification of classical algorithms due to Dijkstra and Prim allows quantum search to replace classical search and leads to more efficient algorithms. In the case of highly asymmetric complete bipartite graphs, simply replacing classical search with quantum search leads to a faster quantum algorithm. A fast quantum algorithm for computing the diameter of a complete graph is also given.
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.
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.
The theory of quantum complexity determines when quantum computers may offer a computational speed-up over classical computers. At present, there are only a few general well-known techniques in the field of quantum computing and finding the problems that are amenable to quantum speedups is a high priority. Nonetheless, efficient quantum algorithms are very limited in number and scope; no real breakthrough has yet been achieved in physical implementations. Most importantly, these algorithms are not still matured adequately to be applied in real quantum computations. The Quantum Processing Unit (QPU) is the processor of quantum computer that is able to do quantum computations. A typical component in QPU is a quantum device that runs quantum algorithms; namely Quantum Algorithm Processing Unit (QAPU). The focus of this study is developing a framework of QAPU and hybrid architecture for classical-quantum algorithms. The framework is used to increase the implementation performance of quantum algorithms. The framework shows a general plan for the architecture of quantum processor which is capable to run the quantum algorithms. In particular, QAPU can be used as a quantum node to design a quantum multicomputer. A uniform platform was proposed for the QAPU used in the present study. At first, the hybrid architecture was designed for the quantum algorithms. Then, the relationship between classical and quantum parts of the hybrid algorithms were extracted and the main stages of the hybrid algorithm were determined. Next, the framework of QAPU was designed and developed. For this purpose, some gates and connections were projected in the framework which could be applied for future quantum algorithms. Furthermore, the framework was set up, implemented and simulated for the existing quantum algorithms on a classic computer. The results were shown that the framework is appropriate for the quantum algorithms and had been mathematically proven. Moreover, a quantum algorithm was designed to solve the shortest paths problem between the two points in a network. Another quantum algorithm for a minimum weight spanning tree in the graph was also designed. These quantum algorithms can be applied to solve routing problem in the quantum networks and the quantum multicomputer. The designed hybrid architecture and framework were tested using the above quantum algorithms and the existing quantum algorithms. Implementing and simulating results are then represented based on the probabilistic hybrid algorithm in the case of linear array, binary heap and quantum search for dense and sparse graphs.