  For quantum computing to become fault tolerant, the underlying quantum bits
must be effectively isolated from the noisy environment. It is well known that
including an electromagnetic bandgap around the qubit operating frequency
improves coherence for superconducting circuits. However, investigations of
bandgaps to other environmental coupling mechanisms remain largely unexplored.
Here we present a method to enhance the coherence of superconducting circuits
by introducing a phononic bandgap around the device operating frequency. The
phononic bandgaps block resonant decay of defect states within the gapped
frequency range, removing the electromagnetic coupling to phonons at the gap
frequencies. We construct a multi-scale model that derives the decrease in the
density of states due to the bandgap and the resulting increase in defect state
$T_1$ times. We demonstrate that emission rates from in-plane defect states can
be suppressed by up to two orders of magnitude. We combine these simulations
with theory for resonators operated in the continuous-wave regime and show that
improvements in quality factors are expected by up to the enhancement in defect
$T_1$ times. Furthermore, we use full master equation simulation to demonstrate
the suppression of qubit energy relaxation even when interacting with 200
defects states. We conclude with an exploration of device implementation
including tradeoffs between fabrication complexity and qubit performance.
