  The key for realizing fault-tolerant quantum computation lies in maintaining
the coherence of all qubits so that high-fidelity and robust quantum
manipulations on them can be achieved. One of the promising approaches is to
use geometric phases in the construction of universal quantum gates, due to
their intrinsic robustness against certain types of local noises. However, due
to limitations in previous implementations, the noise-resilience feature of
nonadiabatic holonomic quantum computation (NHQC) still needs to be improved.
Here, combining with the dynamical correction technique, we propose a general
protocol of universal NHQC with simplified control, which can greatly suppress
the effect of the accompanied X errors, retaining the main merit of geometric
quantum operations. Numerical simulation shows that the performance of our gate
can be much better than previous protocols. Remarkably, when incorporating a
decoherence-free subspace encoding for the collective dephasing noise, our
scheme can also be robust against the involved Z errors. In addition, we also
outline the physical implementation of the protocol that is insensitive to both
X and Z errors. Therefore, our protocol provides a promising strategy for
scalable fault-tolerant quantum computation.
