Quantum Learning with Tunable Loss Functions

Yixian Qiu, Lirandë Pira, Patrick Rebentrost

Abstract: Learning from quantum data presents new challenges, particularly regarding robustness in the presence of real-world noise, data contamination, and outliers. While recent work on Quantum Empirical Risk Minimization (QERM) provides a foundational framework for learning from classical–quantum data, its sensitivity to data anomalies can limit its practical utility. To address this, we introduce a tilted empirical risk objective for quantum learning, in which a tilt parameter controls the tradeoff between standard average-loss minimization and a stronger emphasis on hard or atypical samples. We call this framework Quantum Tilted Empirical Risk Minimization (QTERM). This tunable objective provides a flexible alternative to implicit and explicit regularization strategies, allowing the learner to balance worst-case generalization guarantees against robustness to outlier data. Our contributions are threefold. First, we prove learnability by deriving upper bounds on the sample complexity in the small-tilt regime. Second, we establish new PAC generalization bounds for classical TERM and obtain agnostic learning guarantees for QTERM, enabling hypothesis selection from noisy quantum data. Third, we analyze robustness properties for QTERM, including trace-distance stability under quantum distribution shift, robustness to quantum outliers, and an application to robust quantum hypothesis selection, where tilting improves resilience to contamination. Together, these results identify QTERM as a practical and theoretically grounded framework for robust quantum learning in noisy experimental settings.