This dissertation focuses on two connected areas: quantum computation and quantum control. Two proposals to construct a quantum computer, using nuclear magnetic resonance (NMR) and superconductivity, are introduced. We give details about the modeling, qubit realization, one and two qubit gates and measurement in the language that mathematicians can understand and fill gaps in the original literatures. Two experimental examples using liquid NMR are also presented. Then we proceed to investigate an example of quantum control, that of a magnetometer using quantum feedback. Previous research has shown that feedback makes the measurement robust to an unknown parameter, the number of atoms involved, with the assumption that the feedback is noise free. To evaluate the effect of the feedback noise, we extend the original model by an input noise term. We then compute the steady state performance of the Kalman filter for both the closed-loop and open-loop cases and retrieve the estimation error variances. The results are compared and criteria for evaluating the effects of input noise are obtained. Computations and simulations show that the level of input noise affects the measurement by changing the region where closed loop feedback is beneficial.
This dissertation focuses on two connected areas: quantum computation and quantum control. Two proposals to construct a quantum computer, using nuclear magnetic resonance (NMR) and superconductivity, are introduced. We give details about the modeling, qubit realization, one and two qubit gates and measurement in the language that mathematicians can understand and fill gaps in the original literatures. Two experimental examples using liquid NMR are also presented. Then we proceed to investigate an example of quantum control, that of a magnetometer using quantum feedback. Previous research has shown that feedback makes the measurement robust to an unknown parameter, the number of atoms involved, with the assumption that the feedback is noise free. To evaluate the effect of the feedback noise, we extend the original model by an input noise term. We then compute the steady state performance of the Kalman filter for both the closed-loop and open-loop cases and retrieve the estimation error variances. The results are compared and criteria for evaluating the effects of input noise are obtained. Computations and simulations show that the level of input noise affects the measurement by changing the region where closed loop feedback is beneficial.