Numerical Analysis of Quantum State Evolution on Finite Group Symmetry Spaces: A Conjugacy-Class Approach to Discrete Quantum Dynamics

Michael Nsikan John. (2026). Numerical Analysis of Quantum State Evolution on Finite Group Symmetry Spaces: A Conjugacy-Class Approach to Discrete Quantum Dynamics. Ktrend - International Journal of Physical Sciences (IJPS), 1(1), 1-15. https://doi.org/10.5281/zenodo.20683588 

The study of symmetry remains one of the most important themes in modern mathematical physics, providing a rigorous framework for understanding conservation laws, quantum state transitions, and dynamical invariants. Finite groups offer natural algebraic representations of discrete symmetry systems and have found applications in quantum mechanics, cryptography, coding theory, and computational physics. This paper develops a mathematical and numerical framework for quantum state evolution on finite group symmetry spaces using conjugacy-class dynamics. A finite-dimensional state space is constructed from the elements of a finite group, while a conjugacy-class transition matrix is introduced to model interactions among discrete quantum states. The associated conjugacy-class Laplacian generates a system of ordinary differential equations governing the evolution of the state vector. The model is analyzed using spectral theory, energy methods, and numerical approximation. The Forward Euler method and the classical fourth-order Runge–Kutta method are formulated for the proposed dynamical system. Stability and convergence properties are established theoretically, and a spectral stability theorem is proved for the conjugacy-class Laplacian. Numerical experiments based on the symmetric group S3 illustrate the computational behaviour of the model and confirm the superior accuracy of the Runge–Kutta scheme compared with the Euler approximation. The study establishes a direct connection between finite group theory, mathematical physics, and numerical analysis, thereby providing a structured approach to discrete quantum dynamics generated by algebraic symmetry.