  We autonomously stabilize arbitrary states of a qubit through parametric
modulation of the coupling between a fixed frequency qubit and resonator. The
coupling modulation is achieved with a tunable coupler design, in which the
qubit and the resonator are connected in parallel to a superconducting quantum
interference device. This allows for quasi-static tuning of the qubit-cavity
coupling strength from 12 MHz to more than 300 MHz. Additionally, the coupling
can be dynamically modulated, allowing for single photon exchange in 6 ns.
Qubit coherence times exceeding 20 $\mu$s are maintained over the majority of
the range of tuning, limited primarily by the Purcell effect. The parametric
stabilization technique realized using the tunable coupler involves engineering
the qubit bath through a combination of photon non-conserving sideband
interactions realized by flux modulation, and direct qubit Rabi driving. We
demonstrate that the qubit can be stabilized to arbitrary states on the Bloch
sphere with a worst-case fidelity exceeding 80 %.
