  The overhead to construct a logical qubit from physical qubits rapidly
increases with the decoherence rate. Current superconducting qubits reduce
dissipation due to two-level systems (TLSs) by using large device footprints.
However, this approach provides partial protection, and results in a trade-off
between qubit footprint and dissipation. This work introduces a new platform
using phononics to engineer superconducting qubit-TLS interactions. We realize
a superconducting qubit on a phononic bandgap metamaterial that suppresses
TLS-mediated phonon emission. We use the qubit to probe its thermalization
dynamics with the phonon-engineered TLS bath. Inside the phononic bandgap, we
observe the emergence of non-Markovian qubit dynamics due to the
Purcell-engineered TLS lifetime of 34 $\mu s$. We discuss the implications of
these observations for extending qubit relaxation times through simultaneous
phonon protection and miniaturization.
