Commutator-Residual Quantum Circuits: Mitigating Barren Plateaus with an Exponentially Large Lie Algebra

Barren plateaus, the exponential vanishing of gradient variances in parametrized quantum circuits, are the central obstacle to scaling variational algorithms. We introduce the commutator-residual ansatz, a circuit of two-qubit group commutators whose generators are symmetrically drawn from a $Y$-odd Pauli pool. The pool's dynamical Lie algebra (DLA) is the $Y$-odd sector of $\text{su}(2^n)$, dimension $(4^n-2^n)/2$, and the closed form exposes a commutator-degeneracy trap that independent sampling falls into and the CoPairSite protocol, which pairs anti-commuting generators on one site, eliminates. A cross-driving lock ties every gradient to a single-qubit $Y$ expectation, and this carrier mechanism motivates a conjectured polynomial lower bound in the small-initialization regime, which would explain why an exponentially large DLA need not imply a barren plateau. At $n=16$ the CoPairSite variance exceeds the hardware-efficient ansatz (HEA) by three orders of magnitude and decays with width at a fitted log-log slope near $-1$ against HEA's $-8$; the surviving gradients carry through to faster convergence and lower final errors on ground-state preparation and phase-recognition tasks, and the variance contrast survives depolarizing noise, a global cost, and identity-block initialization.