Quantum-Enhanced Conformal Methods for Multi-Output Uncertainty: A Holistic Exploration and Experimental Analysis

Quantum computing introduces unique forms of randomness arising from measurement processes, gate noise, and hardware imperfections. Ensuring reliable uncertainty quantifi- cation in such quantum-driven or quantum-derived predictions is an emerging challenge. In classical machine learning, conformal prediction has proven to be a robust frame- work for distribution-free uncertainty calibration, often focusing on univariate or low- dimensional outputs. Recent advances have extended conformal methods to handle multi-output or multi-dimensional responses, addressing sophisticated tasks such as time-series, image classification sets, and quantum-generated probability distributions. However, bridging the gap between these powerful conformal frameworks and the high- dimensional, noise-prone distributions typical of quantum measurement scenarios remains largely open. In this paper, we propose a unified approach to harness quantum conformal methods for multi-output distributions, with a particular emphasis on two experimental paradigms: (i) a standard 2-qubit circuit scenario producing a four-dimensional outcome distribution, and (ii) a multi-basis measurement setting that concatenates measurement probabilities in different bases (Z, X, Y) into a twelve-dimensional output space. By combining a multi- output regression model (e.g., random forests) with distributional conformal prediction, we validate coverage and interval-set sizes on both simulated quantum data and multi-basis measurement data. Our results confirm that classical conformal prediction can effectively provide coverage guarantees even when the target probabilities derive from inherently quantum processes. Such synergy opens the door to next-generation quantum-classical hybrid frameworks, providing both improved interpretability and rigorous coverage for quantum machine learning tasks. All codes and full reproducible Colab notebooks are made available at https://github.com/detasar/QECMMOU.