Can the Quantum Fisher Effective Dimension Judge a Hybrid Quantum Neural Network?

Abstract

Hybrid quantum neural networks (QNNs) pair a classical feature extractor with a variational quantum circuit, and the effective dimension of the circuit's quantum Fisher information matrix (QFIM) is widely used to assess such models. This scalar is often interpreted as an indicator of trainability or accuracy, although these readings are rarely tested under controlled conditions. We therefore investigate what it actually measures. We define the effective dimension as the participation ratio of the QFIM diagonal, which lies between one and the parameter count, and we evaluate it across two image datasets, four frozen backbones including a random-projection control, four circuit depths on eight qubits, and three seeds. During training, the effective dimension remains well below the parameter count, and its correlation with test accuracy ranges from -0.72 to +0.87, changing sign across seeds in half of the configurations. Increasing depth activates a larger share of the circuit directions, from 0.38 to 0.91 of the parameter count, while the initialization gradient variance remains between 1e-5 and 1e-4, so no barren plateau (BP) appears at eight qubits. At a fixed circuit, the four backbones share an initial effective dimension of 6 to 9, even though CIFAR-10 accuracy ranges from 23.14% to 86.44%. These results show that the effective dimension characterizes circuit geometry rather than model quality. We conclude with a practical protocol that assigns each diagnostic to the question it can answer.