Implementing robust Holonomic quantum gates using dynamical invariant

Holonomic quantum computing operates quantum systems using berry's phase, or more generally, Aharanov-Anandan phase. It is proved that quantum gates implemented by using these geometric phases are more robust against certain errors. While Holonomic gates have been realized in many systems, extra degrees of freedom are usually required. In this work, we propose a general dynamical invariant for 1 qubit and 2 qubit that is restricted in logical space that enables holonomic quantum computing without inverse engineering. In particular, a Holonomic CNOT gate can be implemented with extremely high fidelity. Moreover, an investigation in the aspect of geometry shows the Hamiltonians used to control the system all have monopole-type gauge field in parameter space, which provides a hidden mathematical structure that contributes to extra robustness.