  The key for realizing fault-tolerant quantum computation lies in maintaining
the coherence of all the qubits so that high-fidelity and robust qubit
manipulations can be achievable. One of the promising approaches is to use
geometric phases in the construction of universal quantum gates, due to the
intrinsic robustness against operational errors, i.e., X errors. However, due
to the implementation limitations, previous schemes for nonadiabatic holonomic
quantum computation (NHQC) do not possess the noise-resilience feature. Here,
combining with the dynamical correction technique, we propose a general
protocol for universal NHQC, which can greatly suppress X errors, retaining the
merit of geometric quantum operations. Numerical simulation shows that our gate
performance can be much better than previous protocols. Remarkably, when
incorporate a minimum resource decoherence-free subspace encoding, our scheme
can also be robust against dephasing error, i.e., the Z error. 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.
