Holonomic Quantum Computing

We present a geometric framework for holonomic quantum computing in which quantum gates arise from global properties of control manifolds rather than fine-tuned dynamical evolution. Quantum states are modeled as complex projective fibers over a classical control manifold, and adiabatic loops induce unitary gates through Berry and Wilczek-Zee holonomy. Within this setting, we introduce Quantum Inner State Manifolds (QISMs) as symplectic fiber bundles equipped with a natural unitary connection governed by the Fubini-Study form. Using the Ambrose-Singer theorem, we show that generic QISMs generate holonomy groups dense in U(N), establishing universality. Fault tolerance emerges from global geometric features, providing a robust geometric foundation for quantum gate design.