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

TL;DR

This work tackles the challenge of exactly propagating quantum systems whose Hamiltonians are linear combinations of Lie-algebra generators by leveraging the Wei–Norman method. It introduces Symdyn, a Python library that automates the Wei–Norman factorization, computes BCH-like relations, and constructs the coupling and decoupled differential equations that govern the time evolution for high-order algebras, including $\textit{SU}(N)$. The authors demonstrate the approach on a system of two time-dependent coupled harmonic oscillators and develop generic Cartan–Weyl bases for $\mathfrak{su}(N)$, applying the framework to $\mathfrak{su}(2)$, $\mathfrak{su}(3)$ and $\mathfrak{su}(4)$ to recover key gates such as Hadamard, $T$, and CNOT, thereby enabling circuit-level quantum computing within a Lie-algebraic setting. The results emphasize the importance of basis choice for obtaining globally valid, tractable solutions and suggest significant potential for analyzing high-dimensional quantum dynamics, optimal control, and gate design with scalable automation.

Abstract

Many significant quantum physical systems are characterized by Hamiltonians expressible as a linear combination of time-independent generators of a closed Lie algebra, $\hat{H}(t)=\sum_{l=1}^{L}η_{l}(t)\hat{g}_{l}$. The Wei-Norman method provides a framework for determining the coefficients of the corresponding time evolution operator in its factorized representation, $\hat{U}(t) = \prod_{l=1}^{L} e^{ Λ_{l}(t)\hat{g}_{l}}$. 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.

Figures (2)

• Figure 1: Representation of the Structure Tensor $\gamma$ and its top transverse matrix $\boldsymbol{\Upsilon}^{1}$.
• Figure 2: Flowchart of the WNM.