Protocols for optimal readout of qubits using a continuous quantum nondemolition measurement

We study how the spontaneous relaxation of a qubit affects a continuous quantum nondemolition 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 nonlinear 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 nonlinear 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 similar to 30 in a time T-1, whereas the nonlinear filter only requires a signal to noise ratio of similar to 18.