Finite-Shot Quantum Federated Learning: Trajectory-Dependent Update Discrepancies and Nonconvex Convergence under Proximal Dynamics

We study quantum federated learning with variational quantum classifiers under finite-shot measurements, heterogeneous clients, multiple local updates, and partial participation. Finite-shot parameter-shift gradients are evaluated along recursively evolving proximal client trajectories, so their effect cannot be represented by simply adding an $O(1/S)$ variance term to a standard proximal federated bound. We therefore analyze the trajectory-dependent discrepancy between the actual federated update and an ideal global-gradient step. At the circuit level, we establish unbiasedness of the parameter-shift estimator, coordinatewise variance $O(1/S)$, and gradient mean-squared error $O(d_q/S)$. At the optimization level, we establish first-order stationarity guarantees for smooth nonconvex objectives with time-varying learning rates and proximal coefficients, without a Polyak--{\L}ojasiewicz condition. The analysis identifies two finite-shot effects: a direct stochastic-gradient residual and an indirect trajectory-mediated residual. It also shows that heterogeneity does not create an independent common-iterate $O(\zeta^2)$ error; instead, it acts through subsequent client drift. For uniform sampling without replacement, we derive an additional correction proportional to $(K-m)/(m(K-1))$ and terminal drift energy, which vanishes under full participation. Experiments confirm near-inverse-shot estimator scaling and illustrate the resulting drift and stationarity behavior.