Parallel Quantum Algorithm for Finding the Consistency of Saaty’s Matrices

Abstract

Features of the quantum systems enable simple calculation of the eigenvalues in uniquely - parallel way. We propose the use of quantum algorithms for the implementation of an iterative method to search for consistency in the Saaty’s matrix of relative judgments, by step by step closing up to the consistent matrix structure. Typically, the matrix of relative judgments is prepared on the basis of the opinion of an expert or group of experts. In practice if is necessary to obtain consistent form of opinions set, but when we want to get the desired level of their consistency we must even in the minimal scope correct them. Criteria of correction are: the minimum number of seats of correcting (the fastest convergence to the consistency) or minimum summary value of alterations. In our proposition we can choose several variants of the iterative corrections as a way of consolidation. The method most often chosen by experts is based on the minimal correction in every iteration. Sometimes we want to make minimal iteration steps. Details of this classical approach are presented in [9].

In this paper we want to support the classical algorithm by the quantum convention and parameters. The measurement realization will be connected with the state transition and reading of the eigenvalue. The superposition procedure will be activated after the transition, what causes the change of the probability of the choice of location of next correction(s). In the aspect of quantum calculations we use quantum vectors, qubits, inner and tensor vectors, linear operators, projectors, gates etc. The resulting effect (for simulation of quantum calculations) concerning the complexity of the calculations is comparable to the classical algorithm.