  We experimentally demonstrate fast and high-fidelity geometric control of a
quantum system with the most brachistochrone method on hybrid spin registers in
diamond. Based on the time-optimal universal geometric control, single
geometric gates with the fidelities over 99.2% on the spin state of
nitrogen-vacancy center are realized with average durations shortened by 74.9%,
comparing with conventional geometric method. The fidelity of the fast
geometric two-qubit gate exceeds 96.5% on the hybrid spin registers. With these
fast high-fidelity gates available, we implement quantum entanglement-enhanced
phase estimation algorithm and demonstrate the Heisenberg quantum limit at
room-temperature. By comparing with the conventional geometric circuit, the
measurement bandwidth and sensitivity is enhanced by 3.5 and 2.9 times. Hence,
our results show that high-fidelity quantum control based on a fast geometric
route will be a versatile tool for broad applications of quantum information
processing in practice.
