Uncertainty-Calibrated Quantum Neural Networks via Bayesian Variational Circuits and Conformal Prediction

Reliable decision-making with quantum neural networks on NISQ hardware requires uncertainty estimates that remain meaningful under finite-shot measurement noise and distribution shift. This study introduces an end-to-end pipeline that combines Bayesian variational quantum circuits with post hoc probability calibration and conformal prediction to produce coverage-controlled prediction sets. Across shot budgets from 128 to 4096 shots, the Bayesian variational circuit improves accuracy from 0.828 to 0.839 at 128 shots and from 0.862 to 0.892 at 4096 shots, while reducing negative log-likelihood from 0.462 to 0.448 at 128 shots and from 0.392 to 0.357 at 4096 shots. Calibration preserves accuracy while further lowering negative log-likelihood to 0.432 at 128 shots and 0.350 at 4096 shots, and reduces expected calibration error from 0.058 to 0.027. Conformalization achieves empirical coverage of 0.952 at miscoverage 0.05, with average set size 1.39 and singleton rate 0.71, and maintains coverage between 0.944 and 0.948 under feature corruption and subpopulation shift, where deterministic accuracy degrades to 0.801 and 0.774, respectively, while the Bayesian model retains 0.842 and 0.816. Ablations identify an operational knee at circuit depth 8, and show diminishing returns beyond 8 posterior samples per prediction, supporting deployment-efficient uncertainty control for quantum learning.