A Hybrid Quantum-Classical Framework for Adaptive AI via Nonlinear Self-Reference

This paper develops a hybrid quantum–classical framework for adaptive AI agents, combin- ing a self-reference-aware quantum evaluation layer with a classical candidate-generation and evolutionary optimization layer. On the quantum side, we introduce a nonlinear, memory- dependent extension of open-system dynamics through St[ρ] and derive key structural proper- ties, including trace preservation, Hermiticity, pointer-basis fixed-point behavior, and practical positivity conditions in bounded-coupling regimes. On the AI-systems side, we define measur- able response metrics (χ2, ζ), introduce a compositional synergy integral Sint, and specify an online-selection plus offline-evolution pipeline. Candidate-dependent evaluation is implemented through semantic embedding and amplitude encoding, so quantum initialization reflects linguis- tic proximity rather than hash collisions. The contribution is framed as a testable theoretical architecture rather than a universal performance claim: χ2 and ζ are structural diagnostics, while semantic-quality gains remain an empirical hypothesis requiring calibration. We also pro- vide implementation-oriented interfaces and a worked compositional example to support staged empirical validation on NISQ-era hardware.