  We study how the spontaneous relaxation of a qubit affects a continuous
quantum non-demolition measurement of the initial state of the qubit. Given
some noisy measurement record $\Psi$, we seek an estimate of whether the qubit
was initially in the ground or excited state. We investigate four different
measurement protocols, three of which use a linear filter (with different
weighting factors) and a fourth which uses a full non-linear filter that gives
the theoretically optimal estimate of the initial state of the qubit. We find
that relaxation of the qubit at rate $1/T_1$ strongly influences the fidelity
of any measurement protocol. To avoid errors due to this decay, the measurement
must be completed in a time that decrease linearly with the desired fidelity
while maintaining an adequate signal to noise ratio. We find that for the
non-linear filter the predicted fidelity, as expected, is always better than
the linear filters and that the fidelity is a monotone increasing function of
the measurement time. For example, to achieve a fidelity of 90%, the box car
linear filter requires a signal to noise ratio of $\sim 30$ in a time $T_1$
whereas the non-linear filter only requires a signal to noise ratio of $\sim
18$.
