  Adiabatic quantum control is a powerful tool for quantum engineering and a
key component in some quantum computation models, where accurate control over
the timing of the involved pulses is not needed. However, the adiabatic
condition requires that the process should be very slow and thus limits its
application in quantum computation, where quantum gates are preferred to be
fast due to the limited coherent times of the quantum systems. Here, we propose
a feasible scheme to implement universal holonomic quantum computation based on
non-Abelian geometric phases with superadiabatic quantum control, where the
adiabatic manipulation is sped up while retaining its robustness against errors
in the timing control. Consolidating the advantages of both strategies, our
proposal is thus both robust and fast. The quantum cavity QED system is adopted
as a typical example to illustrate the merits, where the proposed scheme can be
realized in a tripod configuration by appropriately controlling the pulse
shapes and their relative strength. To demonstrate the distinct performance of
our proposal, we also compare our scheme with the conventional adiabatic
strategy.
