  The readout fidelity of superconducting transmon and Xmon qubits is partially
limited by the qubit energy relaxation through the resonator into the
transmission line, which is also known as the Purcell effect. One way to
suppress this energy relaxation is to employ a filter which impedes microwave
propagation at the qubit frequency. We present semiclassical and quantum
analyses for the bandpass Purcell filter realized by E.\ Jeffrey \textit{et
al}.\ [Phys.\ Rev.\ Lett.\ 112, 190504 (2014)]. For typical experimental
parameters, the bandpass filter suppresses the qubit relaxation rate by up to
two orders of magnitude while maintaining the same measurement rate. We also
show that in the presence of a microwave drive the qubit relaxation rate
further decreases with increasing drive strength.
