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Symdyn: An automated algebraic solution for high-order quantum systems

D. Martínez-Tibaduiza1,*, Vladimir Vargas-Calderón2, J. G. Dueñas3, J. Flórez-Jiménez4, and A. Z. Khoury1

  • 1Instituto de Física, Universidade Federal Fluminense, Avenida Litorânea, 24210-346 Niteroi, RJ, Brazil
  • 2D-Wave Systems Inc., Burnaby, British Columbia, Canada
  • 3Information Analysis Group, Special Jurisdiction for Peace, Cra 7 No 63-44, Bogotá, Colombia
  • 4Lyric, Sunnyvale, California 94087, USA

  • *Contact author: danielmartinezt@gmail.com
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Phys. Rev. Research 8, 013078 – Published 26 January, 2026

DOI: https://doi.org/10.1103/24r3-j9zy

Abstract

Many important quantum systems are characterized by Hamiltonians expressible as a linear combination of time-independent generators of a finite Lie algebra, Ĥ(t)=∑l=1Lηl(t)ĝl. The Wei-Norman framework provides a powerful analytical method for computing the exact time-evolution operator in a factorized form, Û(t)=∏l=1LeΛl(t)ĝl. By exploiting the underlying algebraic structure, the method yields a system of differential equations—the Wei-Norman equations—which determine the time-evolution operator coefficients Λl(t) from the Hamiltonian ones ηl(t). Although robust and formally extendable, applying the method by hand becomes increasingly difficult or even intractable for high-order systems (L≥6). In this work, we introduce symdyn, a Python library that automates the application of this method. Given the commutators between generators, the library symbolically derives the Wei-Norman equations and efficiently computes nested commutators, similarity transformations, and the differential equations that yield Baker-Campbell-Hausdorff-like relations. We validate the library by recovering known results for quantum systems associated with the Lie algebras su(1,1), su(2), sl(2), and so(2,1), and further demonstrate its capabilities by deriving the Wei-Norman equations for a system of two coupled time-dependent harmonic oscillators involving 11 generators. Additionally, we specialize the library to handle the Lie group SU(N) using a generic Cartan-Weyl basis for its algebra, and demonstrate its versatility by deriving the Wei-Norman equations for su(2), su(3), and su(4), including the explicit construction of a set of quantum gates for universal quantum computation.

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Phys. Rev. Research 8, 013078– Published 26 January, 2026
  • Received 31 March 2025
  • Accepted 15 December 2025
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