Pure state tomography with adaptive Pauli measurements

Quantum state tomography provides a key tool for validating and fully exploiting quantum resources. However, current protocols of pure-state informationally-complete (PS-IC) measurement settings generally involve various multi-qubit gates or complex quantum algorithms, which are not practical for large systems. In this study, we present an adaptive approach to <inline-formula><tex-math id="M1">\begin{document}$N$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0037_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0037_M1.png"/></alternatives></inline-formula>-qubit pure-state tomography with Pauli measurements. First, projective measurements on each qubit in the <i>Z</i>-direction were implemented to determine the amplitude of each base of the target state. Then, a set of Pauli measurement settings was recursively deduced by the <i>Z</i>-measurement results, which can be used to determine the phase of each base. The number of required measurement settings is <inline-formula><tex-math id="M2">\begin{document}$O(N)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0037_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0037_M2.png"/></alternatives></inline-formula> for certain quantum states, including cluster and <i>W</i> states. Finally, we numerically verified the feasibility of our strategy by reconstructing a 1-D chain state using a neural network algorithm.