Non-Commutative Rough Hybrid Quantum-Classical Information Geometry Dynamics (NC-HQIG): Unifying Hybrid Inseparability, Quantum Fisher Metrics, Noisy Barren Plateaus, and Advantage-Security Duality via UROA, STCT, and RQIG

Realizing practical quantum advantage across the Noisy Intermediate-Scale Quantum (NISQ) and early fault-tolerant regimes requires moving beyond isolated pure-state quantum mechanics to tightly coupled hybrid quantum-classical architectures. This paper establishes Non-Commutative Rough Hybrid Quantum-Classical Information Geometry Dynamics (NC-HQIG) within the SMA-∞ Topos [1]. By integrating Foundations of Roughness Operator Algebra (ROA) [3], Universal Rough Operator Algebra (UROA) [4], Seonggil Tensor Calculus (STCT) [2], and Rough Quantum Information Geometry (RQIG), we prove that quantum-classical channel composition, Bures–Fisher–Rao natural gradient optimization, noise-induced barren plateaus (NIBPs), and the quantum advantage versus post-quantum cryptographic security duality are topological manifestations of non-commutative manifold curvature and information geodesic routing. Specifically, we formalize hybrid quantum classical interface coupling via the UROA Roughness Operator Γ_R, regulate decoherence driven Quantum Fisher Information Matrix (QFIM) eigenvalue collapse via the STCT Dynamic Torsion Tensor T_STCT, and route variational parameter updates along minimal-entropy RQIG geodesics. A self-contained Python numerical simulation suite empirically validates exponential convergence acceleration under Quantum Natural Gradient (QNG) and quantifies noisy barren plateau suppression.