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International Symposium on Information Technology

A quantum algorithm for minimal spanning tree

Mohammad Reza Soltan Aghaei, Zuriati Ahmad Zukarnain, Ali Mamat, Hishamuddin B Zainuddin

Published Aug 1, 2008• 9 citations
Publisher

Abstract

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.

DOI
10.1109/itsim.2008.4632038

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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 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.
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 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.
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.
It has recently been shown that starting with a classical query algorithm (decision tree) and a guessing algorithm that tries to predict the query answers, we can design a quantum algorithm with query complexity O( √ GT where T is the query complexity of the classical algorithm (depth of the decision tree) and G is the maximum number of wrong answers by the guessing algorithm [ 3 , 14 ]. In this article, we show that, given some constraints on the classical algorithms, this quantum algorithm can be implemented in time Õ (√ GT ). Our algorithm is based on non-binary span programs and their efficient implementation. We conclude that various graph-theoretic problems including bipartiteness, cycle detection, and topological sort can be solved in time O(n 3/2 log 2 n ) and with O(n 3/2 ) quantum queries. Moreover, finding a maximal matching can be solved with O(n 3/2 ) quantum queries in time O(n 3/2 log 2 n ), and maximum bipartite matching can be solved in time O(n 2 log 2 n ).
[Truncated abstract] This thesis is split into three parts, each addressing the overarching theme of applying graph theoretic techniques to classi cation and search problems. Part I explores this theme in the context of quantum algorithms, in particular involving quantum walks, coined by Aharonov et al. in 1993 [1]. As a quantum analogue to classical random walks, quantum walks represent a promising candidate for quantum algorithms providing speed-ups over known classical methods. I will discuss two categories of quantum walk algorithms; quantum search and graph isomorphism. In particular, the e ciency of quantum walk based search algorithms on various families of graphs will be investigated, linking both e ciency of quantum circuit implementation and the complexity of the search problem to the symmetry of the underlying graphs. Several families of graphs are presented that are amenable to efficient quantum walk based searching, together with explicit quantum circuit implementations. In the case of the n-cube, I show that O(p2n) steps of a quanum walk are suffcient to nd a marked sub-d-cube (where d