Geometric Saturation Diagnosis: Identifying the Capacity Ceiling of Parameterized Quantum Circuits via QGT Rank

Choosing circuit depth for parameterized quantum circuits remains largely heuristic: circuits are typically deepened until optimization or hardware constraints bind, with little guidance on when additional layers cease to expand expressive capacity. We introduce *Geometric Saturation Diagnosis* (GSD), a training-free diagnostic that tracks the rank of the quantum geometric tensor (QGT) across depths to identify the saturation depth $L^{\star }$—the point at which the state manifold stops growing. The QGT rank equals the local tangent-space dimension of the parameterized state manifold; its plateau signals that the circuit has reached its geometric expressive ceiling. GSD further enables layer-wise capacity profiling via per-layer rank increments and parameter-direction importance ranking via the QGT singular value spectrum. Across hardware-efficient and Hamiltonian-inspired ans\"atze ($N=4$–$10$, TFIM and Heisenberg models), saturation depth satisfies the $\Omega (2^N/N)$ parameter-budget lower bound with decreasing overhead, $L^{\star }$ matches the clean energy of much deeper circuits with substantially fewer parameters, and noise accumulation grows monotonically beyond saturation—identifying $L^{\star }$ as a practical operating point that balances geometric capacity against noise. These results establish QGT rank growth as a principled, geometry-based diagnostic for circuit architecture selection.