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 $\ensuremath{\Psi}$, 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∕T1 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 $\ensuremath{\sim}30$ in a time
T1, whereas the nonlinear filter only requires a signal to noise ratio of $\ensuremath{\sim}18$.
As quantum systems expand in size and complexity, manual qubit characterization and gate optimization will be a non-scalable and time-consuming venture. Physical qubits have to be carefully calibrated because quantum processors are very sensitive to the external environment, with control hardware parameters slowly drifting during operation, affecting gate fidelity. Currently, existing calibration techniques require complex and lengthy measurements to independently control the different parameters of each gate and are unscalable to large quantum systems. Therefore, fully automated protocols with the desired functionalities are required to speed up the calibration process. This paper aims to propose single-qubit calibration of superconducting qubits under continuous weak measurements from a real physical experimental settings point of view. We propose a real-time optimal estimator of qubit states, which utilizes weak measurements and Bayesian optimization to find the optimal control pulses for gate design. Our numerical results demonstrate a significant reduction in the calibration process, obtaining a high gate fidelity. Using the proposed estimator we estimated the qubit state with and without measurement noise and the estimation error between the qubit state and the estimator state is less than 0.02. With this setup, we drive an approximated pi pulse with final fidelity of 0.9928. This shows that our proposed strategy is robust against the presence of measurement and environmental noise and can also be applicable for the calibration of many other quantum computation technologies.
We study quantum tomography based on a stochastic continuous-time measurement record obtained from a probe field collectively interacting with an ensemble of identically prepared systems. In comparison to previous studies, we consider here the case in which the measurement-induced backaction has a non-negligible effect on the dynamical evolution of the ensemble. We formulate a maximum likelihood estimate for the initial quantum state given only a single instance of the continuous diffusive measurement record. We apply our estimator to the simplest problem: state tomography of a single pure qubit, which, during the course of the measurement, is also subjected to dynamical control. We identify a regime where the many-body system is well approximated at all times by a separable pure spin coherent state, whose Bloch vector undergoes a conditional stochastic evolution. We simulate the results of our estimator and show that we can achieve close to the upper bound of fidelity set by the optimal generalized measurement. This estimate is compared to, and significantly outperforms, an equivalent estimator that ignores measurement backaction.