  Bosonic codes offer a hardware-efficient strategy for quantum error
correction by redundantly encoding quantum information in the large Hilbert
space of a harmonic oscillator. However, experimental realizations of these
codes are often limited by ancilla errors propagating to the encoded logical
qubit during syndrome measurements. The Kerr-cat qubit has been proposed as an
ancilla for these codes due to its theoretically-exponential noise bias, which
would enable fault-tolerant error syndrome measurements, but the coupling
required to perform these syndrome measurements has not yet been demonstrated.
In this work, we experimentally realize driven parametric coupling of a
Kerr-cat qubit to a high-quality-factor microwave cavity and demonstrate a gate
set enabling universal quantum control of the cavity. We measure the
decoherence of the cavity in the presence of the Kerr-cat and discover excess
dephasing due to heating of the Kerr-cat to excited states. By engineering
frequency-selective dissipation to counteract this heating, we are able to
eliminate this dephasing, thereby demonstrating a high on-off ratio of control.
Our results pave the way toward using the Kerr-cat to fault-tolerantly measure
error syndromes of bosonic codes.
