On Quantum Generation of Random Sequences as the Basis for Constructing Bicyclic Orthogonal Matrices

Abstract

The paper identifies three main approaches to obtaining Hadamard matrices: search using combinatorial methods, calculation with control based on the theory of dynamical systems and the construction of matrices of fixed structures. For the search and construction of Hadamard matrices, the main tool of the source material is the generation of random sequences. The issues of fixing the structures of Hadamard matrices in the form of a bicyclic construction are considered. To obtain such high-order matrices, important procedures are the generation, filtering and selection of such pairs of sequences that an orthogonal matrix of order n could be constructed from the cyclic matrices of order n/2 obtained on their basis. There is a significant influence of the quality of generated random sequences on the construction time of bicyclic matrices. The results of the first experiments with 1 million random sequences of length 100 generated on a quantum generator based on the interference effect of laser pulses with a random phase are presented. In particular, previously unknown Hadamard matrices of orders up to 100 bicyclic structures and maximum determinant matrices on non-Hadamard orders were obtained in a computer experiment.