Sculpting Quantum Landscapes: Fubini-Study Metric Conditioning for Geometry-Aware Learning in Parameterized Quantum Circuits

We introduce a novel meta-learning framework that explicitly conditions the Fubini-Study (FS) metric tensor of Parameterized Quantum Circuits (PQCs) to address the challenges of barren plateaus in Variational Quantum Algorithms (VQAs). While VQAs offer a promising paradigm for near-term quantum computing, their practical efficacy is often limited by poorly conditioned optimization landscapes. Our theoretical analysis establishes that the logarithmic condition number of the FS metric is a pivotal geometric quantity governing trainability, optimization dynamics, and generalization bounds. We propose Sculpture, a classical meta-model that learns to generate data-dependent PQC initializations by minimizing logarithmic condition number, thereby promoting an isotropic and tractable parameter space. Empirical results demonstrate that meta-training successfully reduces logarithmic condition number from approximately 1.47 to 0.64, achieved by substantially increasing the minimum eigenvalue and slightly decreasing the maximum eigenvalue, effectively mitigating barren plateaus. This learned conditioning robustly generalizes to unseen data, consistently yielding well-conditioned PQC initializations. In a downstream hybrid quantum-classical classification task on the Kaggle diabetes dataset, we demonstrate that increasing the meta-scaling coefficient significantly enhances training efficiency, resulting in faster convergence, lower training loss, and reduced gradient norms. Crucially, higher meta-scaling coefficient values consistently yield superior generalization performance, with test accuracy sharply increasing from around 0.68 to over 0.78. Our findings confirm that sculpting the quantum landscape via meta-learning provides a principled form of geometric regularization, fundamentally improving the trainability, optimization dynamics, and generalization capabilities of PQCs. This work offers a powerful avenue for developing more robust and efficient VQAs.