Quantum 10, 2119 (2026). https://doi.org/10.22331/q-2026-05-29-2119 Dynamical Lie algebras (DLAs) have emerged as a valuable tool in the study of parameterized quantum circuits, helping to characterize both their expressiveness and trainability. In particular, the absence or presence of barren plateaus (BPs) – flat regions in parameter space that prevent the efficient training of variational quantum algorithms – has recently been shown to be intimately related to quantities derived from the asso