This paper presents a ground-breaking framework—Neuro-Quantum Iterative Algorithm (NQIA)—that unifies neural meta-learning with quantum-inspired operator dynamics to solve fixed point problems in abstract functional spaces. Unlike conventional schemes limited by strict contractiveness or static update rules, NQIA introduces an adaptive, layered convergence mechanism governed by novel neuro-symbolic residual control and unitary evolution protocols. We establish original theorems that rigorously demonstrate strong convergence in Hilbert spaces, weak convergence in Banach spaces, and residual minimization stability under hybrid topological conditions. Specifically, the Layered Convergence Theorem ensures robust progression of iterates under firmly non-expansive mappings; the Optimal Ω-Selection Theorem derives the spectral-radius minimizing learning rate from eigenvalue analysis; the Residual Stability Theorem introduces a neurosymbolic energy-based correction scheme that guarantees contracting and convergence; and the Topological Residual-Contraction Theorem extends the approach to reflexive Banach spaces equipped with weak topology. This research opens novel directions in the theory and practice of fixed-point algorithms, transcending classical assumptions through hybrid dynamical learning.