Trainability as differential information transmission in variational quantum circuits

Abstract Understanding the origins of trainability in variational quantum circuits remains one of the central challenges in quantum machine learning. Existing approaches based on dynamical Lie algebras characterize the reachable orbit generated by a variational ansatz and provide important information about controllability and expressivity. However, optimization is inherently a local differential process, and the geometric mechanisms governing the transmission of optimization information remain poorly understood. In this work, we develop a differential-geometric framework for trainability based on the variational Jacobian. The Jacobian is interpreted as the differential map connecting the parameter manifold with the tangent space of the reachable orbit. \textcolor{blue}{We derive an exact modal decomposition of gradient generation in terms of Jacobian singular directions and task-dependent modal information, showing that gradient suppression may arise from spectral attenuation, from a reduction of the task-gradient magnitude, from directional misalignment between the task gradient and the Jacobian-accessible subspace, or from combinations of these effects.} We further establish an explicit connection between the variational Jacobian and the underlying dynamical Lie algebra through effective commutator directions. \textcolor{blue}{In the numerical settings considered here, coarse task-independent spectral diagnostics such as the Jacobian rank, the largest singular value, and the total spectral weight exhibit little correlation with optimization performance, whereas normalized directional alignment and task-weighted spectral transmission retain significant predictive associations.} \textcolor{blue}{Finally, we show that Jacobian degeneration suppresses the task-Hessian component transmitted through the variational map, while the additional second-order map contribution need not vanish.} \textcolor{blue}{Taken together, these results provide a geometric framework in which local trainability can be viewed as emerging from the interaction between Lie-algebraic structure, Jacobian geometry, and the learning task, and as a problem of differential information transmission rather than accessibility alone.}