  Quantum computation with quantum gates induced by geometric phases is
regarded as a promising strategy in fault tolerant quantum computation, due to
its robustness against operational noises. However, because of the parametric
restriction of previous schemes, the main robust advantage of holonomic quantum
gates is smeared. Here, we experimentally demonstrate a solution scheme,
demonstrating nonadiabatic holonomic single qubit quantum gates with optimal
control in a trapped Yb ion based on three level systems with resonant drives,
which also hold the advantages of fast evolution and convenient implementation.
Compared with corresponding previous geometric gates and conventional dynamic
gates, the superiority of our scheme is that it is more robust against control
amplitude errors, which is confirmed by the measured gate infidelity through
both quantum process tomography and random benchmarking methods. In addition,
we also outline that nontrivial two qubit holonomic gates can also be realized
within current experimental technologies. Therefore, our experiment validates
the feasibility for this robust and fast holonomic quantum computation
strategy.
