  Quantum gates induced by geometric phases are intrinsically robust against
noise due to their global properties of the evolution paths. Compared to
conventional nonadiabatic geometric quantum computation (NGQC), the recently
proposed nonadiabatic noncyclic geometric quantum computation (NNGQC) works in
a faster fashion, while still remaining the robust feature of the geometric
operations. Here, we experimentally implement the NNGQC in a single trapped
ultracold $^{40}$Ca$^{+}$ ion for verifying the noise-resilient and fast
feature. By performing unitary operations under imperfect conditions, we
witness the advantages of the NNGQC with measured fidelities by quantum process
tomography in comparison with other two quantum gates by conventional NGQC and
by straightforwardly dynamical evolution. Our results provide the first
evidence confirming the possibility of accelerated quantum information
processing with limited systematic errors even in the imperfect situation.
