Quantum Natural Gradient vs. Adam Optimizer in Variational Quantum Classifiers: Crossover Analysis and Information Acquisition Efficiency

Abstract

Variational Quantum Circuits (VQCs) represent a central paradigm in near-term quantum machine learning, yet the comparative optimisation dynamics of quantum-aware and classical optimisers remain insufficiently characterised in realistic multi-class settings. We present a systematic empirical study comparing the Quantum Natural Gradient (QNG) optimizer against Adam within a VQC trained for ten-class digit recognition, employing eight qubits, three variational layers with RY-RZ-RX gate sequences, circular CX entanglement, and data re-uploading—yielding 72 trainable quantum parameters augmented by a classical linear readout head. A diagonal Quantum Fisher Information Matrix (QFIM) estimated via the parameter-shift fidelity metric underpins QNG, with a lazy update scheme (every five gradient steps) to equalise computational cost with Adam. Over 1,000 training epochs, Adam achieves 95% test accuracy while QNG achieves 92%, with QNG demonstrating markedly superior early convergence. Crossover analysis across four encoding-overlap bins confirms that QNG outperforms Adam exclusively in the low-overlap regime (c_i < 0.54), consistent with the theoretical predictions of Kimura and Mitarai [10]. Information acquisition efficiency measurements reveal qualitatively opposite scaling behaviours: Adam’s Fisher-empirical g_i scales as c_i^−2.19 , whereas QNG’s QFIM-fidelity g_i is near scale-invariant (c_i^0.02). These findings provide actionable optimizer selection criteria for practical VQC deployments and offer the first empirical validation of the Kimura-Mitarai efficiency framework beyond the quantum phase estimation setting