Analytical Quantum State Approximation with Pauli Operators Under Trace Norm and Fidelity

Building upon the theoretical foundation of utilizing the eigenvectors of Pauli matrices for single-qubit quantum state preparation, we employ a set of easily preparable quantum states constructed from the eigenvectors of Pauli matrices and arbitrary real quantum logic gates. Using fidelity and trace norm as tools, we achieve the approximation of single-qubit quantum states. For different scenarios, we present complete exact solutions and analytical expressions for the optimal approximation. Furthermore, during the research, we investigated the comparative effectiveness of different distance metrics within the set B_α''' for quantum state approximation. It was found that trace norm can approximate target state more easily than fidelity. This discovery significantly aids in the selection of measurement tools for quantum state approximation tasks.