Home / Current Issue / Paper 1718493
Trigonometric Near-Rings as a Unified Algebraic Framework for Quantum Computation
Subject area: Science,Engineering and Technology · Area of research: Maths and physics
DOI: https://doi.org/10.64388/IREV9I11-1718493
Abstract
We introduce a unified algebraic framework for quantum computation based on finite trigonometric near-rings. Motivated by unification ideas in physics, the proposed structure connects trigonometric functions, non-commutative algebra, and quantum gate operations within a single finite system. By constructing near-rings generated by discrete phase operators and the Hadamard operator, we show that quantum phase interference and basis transformations naturally give rise to non-commutative near-ring structures. The developed framework generalizes earlier Heisenberg-type matrix near-rings to a trigonometric setting, providing a discrete algebraic model for quantum circuits involving phase and Clifford gates. Explicit finite examples including a non-commutative near-ring of order sixteen are presented to illustrate the theory. This approach offers a conceptually simple yet mathematically rigorous bridge between trigonometry and quantum computation and highlights the relevance of near-ring theory as a foundational tool for modeling finite quantum systems.
Keywords
Trigonometric Near-Rings, Quantum Computing; Non-Commutative Algebra, Hadamard Gate, Phase Gates, Finite Near-Rings, Heisenberg Near-Rings, Quantum Circuits, Algebraic Unification.
References
[1] G. Pilz, Near-Rings: The Theory and Its Applications, Amsterdam, Netherlands: North-Holland Publishing Company, 1983.
[2] J. D. P. Meldrum, Near-Rings and Their Links with Groups, London, UK: Pitman Publishing, 1985.
[3] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.
[4] Brian Hall, Quantum Theory for Mathematicians, Springer, 2013.
[5] Antoni Zygmund, Trigonometric Series, Cambridge University Press, 2002.
[6] I. Bengtsson and K. Życzkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement, Cambridge University Press, 2006.
[7] “Finite Near-Rings Using Trigonometric Solutions of Differential Equations,” International Journal of Innovative Research and Applications, 2026. This work introduces finite trigonometric near-rings using sine and cosine solution spaces.
[8] S. Uma, R. Balakrishnan, and T. Tamizh Chelvam, “α₁, α₂ Near-Rings,” International Journal of Algebra, vol. 4, no. 2, pp. 71–79, 2010.
[9] “An Introduction to the Theory of Matrix Near-Rings,” discussing algebraic extensions of near-rings and matrix representations relevant for computational models.
[10] A. Smoktunowicz and L. Vendramin, “On Skew Braces,” arXiv:1705.06958, 2017; discusses links among rings, near-rings, and algebraic structures related to Yang–Baxter theory.
How to cite this paper
@article{1718493,
author = {Dr. K V Rama Rao, M. Sriniva, T. V. Ram Babu, K. Anuradh},
title = {Trigonometric Near-Rings as a Unified Algebraic Framework for Quantum Computation},
journal = {Iconic Research And Engineering Journals},
year = {2026},
volume = {9},
number = {11},
pages = {4916-4918},
issn = {2456-8880},
url = {https://www.irejournals.com/formatedpaper/1718493.pdf},
abstract = {We introduce a unified algebraic framework for quantum computation based on finite trigonometric near-rings. Motivated by unification ideas in physics, the proposed structure connects trigonometric functions, non-commutative algebra, and quantum gate operations within a single finite system. By constructing near-rings generated by discrete phase operators and the Hadamard operator, we show that quantum phase interference and basis transformations naturally give rise to non-commutative near-ring structures. The developed framework generalizes earlier Heisenberg-type matrix near-rings to a trigonometric setting, providing a discrete algebraic model for quantum circuits involving phase and Clifford gates. Explicit finite examples including a non-commutative near-ring of order sixteen are presented to illustrate the theory. This approach offers a conceptually simple yet mathematically rigorous bridge between trigonometry and quantum computation and highlights the relevance of near-ring theory as a foundational tool for modeling finite quantum systems.},
keywords = {Trigonometric Near-Rings, Quantum Computing; Non-Commutative Algebra, Hadamard Gate, Phase Gates, Finite Near-Rings, Heisenberg Near-Rings, Quantum Circuits, Algebraic Unification.},
month = {May},
doi = {https://doi.org/10.64388/IREV9I11-1718493}
}