  Increasing the fidelity of single-qubit gates requires a combination of
faster pulses and increased qubit coherence. However, with resonant qubit drive
via a capacitively coupled port, these two objectives are mutually
contradictory, as higher qubit quality factor requires a weaker coupling,
necessitating longer pulses for the same applied power. Increasing drive power,
on the other hand, can heat the qubit's environment and degrade coherence. In
this work, by using the inherent non-linearity of the transmon qubit, we
circumvent this issue by introducing a new parametric driving scheme to perform
single-qubit control. Specifically, we achieve rapid gate speed by pumping the
transmon's native Kerr term at approximately one third of the qubit's resonant
frequency. Given that transmons typically operate within a fairly narrow range
of anharmonicity, this technique is applicable to all transmons. In both theory
and experiment, we show that the Rabi rate of the process is proportional to
applied drive amplitude cubed, allowing for rapid gate speed with only modest
increases in applied power. In addition, we demonstrate that filtering can be
used to protect the qubit's coherence while performing rapid gates, and present
theoretical calculations indicating that decay due to multi-photon losses, even
in very strongly coupled drive lines, will not limit qubit lifetime. We
demonstrate $\pi/2$ pulses as short as tens of nanoseconds with fidelity as
high as 99.7\%, limited by the modest coherence of our transmon. We also
present calculations indicating that this technique could reduce cryostat
heating for fast gates, a vital requirement for large-scale quantum computers.
