  In circuit quantum electrodynamics, measuring the state of a superconducting
qubit introduces a loss channel which can enhance spontaneous emission through
the Purcell effect, thus decreasing qubit lifetime. This decay can be mitigated
by performing the measurement through a Purcell filter which forbids signal
propagation at the qubit transition frequency. If the filter is also
well-matched at the readout cavity frequency, it will protect the qubit from
decoherence channels without sacrificing measurement speed. We propose and
analyze design for a mechanical Purcell filter, which we also fabricate and
characterize at room temperature. The filter is comprised of an array of
nanomechanical resonators in thin-film lithium niobate, connected in a ladder
topology, with series and parallel resonances arranged to produce a bandpass
response. The modest footprint, steep band edges, and absence of cross-talk in
these filters make them a novel and appealing alternative to analogous
electromagnetic versions currently used in microwave quantum machines.
