Exact Algebraic Reduction of Quantum Neural Circuits: Observable-Dependent Renormalization and Curvature-Based Reliability Control

This conference presentation introduces an algebraic framework for the exact reduction of quantum neural circuits through observable-dependent renormalization and geometric reliability control. The proposed approach addresses fundamental challenges in quantum machine learning, including circuit complexity, fragmented mathematical representations, training stability, and the reliability of generative models. The framework represents parameterized quantum circuits as quantum dynamical semigroups and constructs an observable-dependent symmetry subalgebra using dynamical Lie algebra closure. This construction enables the derivation of reduced effective models that preserve designated observables exactly while maintaining complete positivity and trace preservation. The presentation further investigates the geometric structure of reduced quantum models through quantum Fisher information, Berry curvature, and context-dependent attention metrics. Geometric regularization is proposed to improve training stability and address barren plateaus, while curvature-based reliability monitoring provides a mathematical approach to detecting deviations of generated outputs from learned data manifolds. The proposed methodology offers potential applications in resource-efficient quantum machine learning, hybrid quantum-classical computing, model robustness, and AI reliability. Future research directions include hardware-aware algebraic reduction, continuous-variable quantum systems, and curvature-based monitoring for trustworthy artificial intelligence.