Quantum Logical Bioinformatics: Genetic Alphabet of Four Hadamard Unitary Operators, and Cyclic Groups

This article is devoted to algebraic-operator modeling of genetic informatics systems providing the inheritance of logical structures and functional algorithms. Formalisms of quantum mechanics and quantum information science are used. The author describes the identification of algebraic-operator alphabets due to algebraic analysis of structures of molecular genetic coding, which serve as the basis for developing cyclic-spiral bioinformatics. This algebraic bioinformatics exists in parallels with established biochemical informatics, which is based on the alphabets of four nucleotides in DNA and RNA and which describes the inheritance of amino acid sequences in proteins. Algebraic bioinformatics expands the possibilities for modeling the inheritance of cyclic and spiral algorithmic structures in living organisms using the formalisms of quantum information science, unitary Hadamard operators, hypercomplex dual numbers, the screw calculus, and Fibonacci matrices. The author presents genetic algebraic-operator foundations for the inheritance and modeling of a variety of geometrically regular biological forms (mollusk shells, phyllotaxis configurations, etc.). Based on the obtained results, a paradigm of code algebraic-operator Darwinism is formulated, according to which natural selection and the inheritance of the most survival-promoting code combinations of alphabetic operators of algebraic bioinformatics play an important role in evolution. This paradigm explains the rapid evolution of organisms as follows: complex tissues are formed not so much by the emergence of new genes, but by algebraic bioinformatics operators that alter the ways existing genes are used, being linked to electromagnetic waves and resonance mechanisms. The role of bioantenna arrays in the energy-informational evolution of living organisms is discussed, including the proposed biological role of hopfion crystals, which have structural analogies with the DNA double helix, as noted by the author.