  Extracting information from quantum devices has long been a crucial problem
in the field of quantum mechanics. By performing elaborate measurements,
quantum state tomography, an important and fundamental tool in quantum science
and technology, can be used to determine unknown quantum states completely. In
this study, we explore methods to determine multi-qubit pure quantum states
uniquely and directly. Two adaptive protocols are proposed, with their
respective quantum circuits. Herein, two or three observables are sufficient,
while the number of measurement outcomes is either the same as or fewer than
those in existing methods. Additionally, experiments on the IBM 5-qubit quantum
computer, as well as numerical investigations, demonstrate the feasibility of
the proposed protocols.
