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Abstract
This thesis develops new algorithms and investigates optimisation in quantum detector tomography (QDT) and quantum process tomography (QPT).
QDT is a fundamental technique for calibrating quantum devices and performing quantum engineering tasks. We design optimal probe states based on the minimum upper bound of the mean squared error (UMSE) and the maximum robustness. We establish the lower bounds of the UMSE and the condition number for the probe states, and provide concrete examples that can achieve these lower bounds. In order to enhance the estimation precision, we also propose a two-step adaptive QDT and present a sufficient condition on when the infidelity scales $ O(1/{N}) $ where $ N $ is the number of state copies.
We then utilize regularization to improve the QDT accuracy whenever the probe states are informationally complete or informationally incomplete. We discuss different regularization forms and prove the mean squared error scales as $ O(1/{N}) $ or tends to a constant with $ N $ state copies under the static assumption. We also characterize the ideal best regularization for the identifiable parameters.
QPT is a critical task for characterizing the dynamics of quantum systems and achieving precise quantum control. We firstly study the identification of time-varying decoherence rates for open quantum systems. We expand the unknown decoherence rates into Fourier series and take the expansion coefficients as optimisation variables. We then convert it into a minimax problem and apply sequential linear programming technique to solve it.
For general QPT, we propose a two-stage solution (TSS) for both trace-preserving and non-trace-preserving QPT. Using structure simplification, our algorithm has $O(MLd^2) $ computational complexity where $d$ is the dimension of the quantum system and $ M $, $ L $ are the type numbers of different input states and measurement operators, respectively. We establish an analytical error upper bound and then design the optimal input states and the optimal measurement operators, which are both based on minimizing the error upper bound and maximizing the robustness characterized by the condition number.
A quantum optical experiment test shows that a suitable regularization form can reach a lower mean squared error in QDT and the testing on IBM quantum machine demonstrates the effectiveness of our TSS algorithm for QPT.
