Depth-dependent separability growth in variational quantum neural networks and its empirical coupling to a parameter-sensitivity statistic

Abstract

Understanding how depth-dependent quantum evolution reshapes the geometric organization of data representations in quantum neural networks (QNNs) remains a central question in quantum machine learning. We study a hardware-efficient variational QNN—single-qubit RY/ RZ rotations per layer followed by a nearest-neighbor CNOT entangling ring, with circuit depth n ranging from 3 to 12 layers—and ask how its data representation geometry reorganizes as n increases. We report two findings across MNIST and BloodMNIST (15 independent random seeds for amplitude encoding: 5 MNIST, 10 BloodMNIST), with a BloodMNIST angle-encoding control (10 further seed-runs, 25 in total) used to check the second finding’s recurrence under a different encoding strategy. First, class separability shows power-law growth within the tested depth window, S(n)∝nb, where b=δ−ε captures the balance between inter-class expansion and intra-class diffusion; this description holds throughout the tested range for MNIST, whereas for BloodMNIST the growth is better described by a model approaching a depth-independent ceiling within the same range, with b retained as a common cross-dataset summary exponent. Second, we measure an auxiliary parameter-perturbation sensitivity statistic and find that its depth-trend exponent γ shows a dataset-level association with b via b≈kdata(1−|γ|). We explicitly tested whether this statistic tracks the textbook state-based quantum Fisher information by computing the latter rigorously for the same trained circuits, and found that it does not; we also find that the b– γ association is weak and not statistically significant within either dataset alone. We therefore report this coupling descriptively, as an open dataset-level pattern rather than an established physical law. To our knowledge, this is the first report of this depth-dependent separability-growth pattern in QNNs together with a practical, predictive heuristic for depth selection derived from it, using each dataset’s own functional form. In the tested depth regime ( n∈[3,12]), cross-seed coefficients of variation for b remain below 10% for amplitude encoding (MNIST: 8.4%, BloodMNIST: 8.1%). Separability fits satisfy mean R2⩾0.773. These results provide a practical, transparently scoped empirical characterization of QNN representation geometry, restricted to the tested datasets, architectures, and depth window.