  A controlled evolution generated by nonlinear interactions is required to
perform full manipulation of a quantum system, and such control is only
coherent when the rate of nonlinearity is large compared to the rate of
decoherence. As a result, engineered quantum systems typically rely on a bare
nonlinearity much stronger than all decoherence rates, and this hierarchy is
usually assumed to be necessary. In this work, we challenge this assumption by
demonstrating the universal control of a quantum system where the relevant rate
of bare nonlinear interaction is comparable to the fastest rate of decoherence.
We do this by introducing a novel noise-resilient protocol for the universal
quantum control of a nearly-harmonic oscillator that takes advantage of an
in-situ enhanced nonlinearity instead of harnessing a bare nonlinearity. Our
experiment consists of a high quality-factor microwave cavity with
weak-dispersive coupling to a much lower quality superconducting qubit. By
using strong drives to temporarily excite the oscillator, we realize an
amplified three-wave-mixing interaction, achieving typical operation speeds
over an order of magnitude faster than expected from the bare dispersive
coupling. Our demonstrations include preparation of a single-photon state with
$98\pm 1(\%)$ fidelity and preparation of squeezed vacuum with a squeezing
level of $11.1$ dB, the largest intracavity squeezing reported in the microwave
regime. Finally, we also demonstrate fast measurement-free preparation of
logical states for the binomial and Gottesman-Kitaev-Preskill (GKP) quantum
error-correcting codes.
