Geometric quantum gates of non-closed paths under counterdiabatic driving

We propose a high-fidelity quantum control framework based on the quasi-topological number (νqua), which extends the traditional Chern number to characterize geometric responses in non-closed paths. By introducing an Adiabatic Gauge Potential that dynamically suppresses non-adiabatic transitions and reconstructs path curvature, we demonstrate that νqua—a relative homotopy invariant of compact manifolds in parameter space—quantifies the robustness of geometric phases during open-path quantum evolution. This integer invariant ensures gauge-invariant suppression of decoherence errors arising from dynamical phase coupling. Numerical simulations in the Kitaev superconducting chain and 2D transverse field Ising model confirm that our protocol achieves high quantum gate fidelity. We bridge geometric quantum control with topological protection, offering a universal approach to obtain high-fidelity and robust quantum gates.