Calibration of Variational Quantum Classifiers Under Depolarizing Noise: Expected Calibration Error, Ansatz Expressibility, and Post-Hoc Temperature Scaling

Abstract

Variational Quantum Classifiers (VQCs) have emerged as prime candidates for machine learning on NISQ systems. It stands to reason that the same depolarizing noise that drives quantum states toward the maximally mixed state would also alleviate the overconfidence of VQCs. This paper tests that hypothesis through an empirical study on three datasets at six noise levels, validated across ten random seeds and supported by a formal analysis of how the depolarizing channel contracts the measured logits. The primary finding is that depolarizing noise does not reduce overconfidence in VQCs that remain in the learnable regime: expected calibration error (ECE) never decreases with noise on any dataset, and on the lowest-variance dataset it increases significantly (Wilcoxon signed-rank p < 0.005 over ten seeds). We derive why the optimizer compensates for the channel and confirm the mechanism through a confidence-trajectory experiment. A secondary finding is that a less expressive ansatz can appear well-calibrated only because it collapses to a degenerate solution, demonstrating that ECE must always be reported alongside accuracy. Any effect of noise on accuracy is small and seed-dependent, and it is decoupled from calibration. Post-hoc temperature scaling reduces VQC ECE by 64 to 77 percent across all datasets and is the recommended calibration method for NISQ-era classifiers.