$\texttt{Symdyn}$: an automated algebraic solution for high-order quantum systems

• URL: http://arxiv.org/abs/2503.22061v2
• Date: Mon, 07 Apr 2025 16:33:13 GMT
• Title: $\texttt{Symdyn}$: an automated algebraic solution for high-order quantum systems
• Authors: D. Martínez-Tibaduiza, Vladimir Vargas-Calderón, J. G. Dueñas, J. Flórez-Jiménez, A. Z. Khoury,
• Abstract summary: This work introduces $textttSymdyn$, a Python library that automates the application of the Wei-Norman method.<n>The library efficiently computes similarity transformations and the nonlinear differential equations intrinsic to derive Baker-Campbell-Hausdorff-like relations.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Many significant quantum physical systems are characterized by Hamiltonians expressible as a linear combination of time-independent generators of a closed Lie algebra, $\hat{H}(t)=\sum_{l=1}^{L}\eta_{l}(t)\hat{g}_{l}$. The Wei-Norman method provides a framework for determining the coefficients of the corresponding time evolution operator in its factorized representation, $\hat{U}(t) = \prod_{l=1}^{L} e^{ \Lambda_{l}(t)\hat{g}_{l}}$. This work introduces $\texttt{Symdyn}$, a Python library that automates the application of this method. The library efficiently computes similarity transformations and the nonlinear differential equations intrinsic to derive Baker-Campbell-Hausdorff-like relations and the time evolution of high-order quantum systems ($L\geq 6$). We demonstrate its robustness by deriving the time evolution operator for a system of two time-dependent coupled harmonic oscillators. Additionally, we specialize the library to the Lie group $\textit{SU}(N)$, showing its versatility with $\textit{SU}(2)$, $\textit{SU}(3)$ and $\textit{SU}(4)$ examples, relevant to quantum computing.

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