PauLie: Fast Classification of Pauli Dynamical Lie Algebras

The dynamical Lie algebra (DLA) governs the controllability, expressibility, and simulation complexity of a quantum system. Explicitly computing it has been a major computational bottleneck: brute-force Lie closure scales exponentially in the number of qubits $n$. Many applications, however, consult only the isomorphism type of the DLA. We introduce PauLie, an open-source framework that decides this isomorphism type for DLAs generated by arbitrary Pauli strings, building on the anticommutation-graph reduction of Aguilar et al. PauLie runs in $O(n\mathrm{\mid }\mathcal{G}\mathrm{\mid }\max (n,\mathrm{\mid }\mathcal{G}\mathrm{\mid }))$ time, where $\mathrm{\mid }\mathcal{G}\mathrm{\mid }$ is the number of generators, turning DLA classification into a routine preprocessing step. We demonstrate its use as a structural oracle for routing Lie-algebraic simulation and Cartan decomposition, diagnosing barren plateaus in variational quantum algorithms, and engineering universal Pauli string generator sets with optimal generation rate.