  We examine the problem of finding the minimum number of Pauli measurements
needed to uniquely determine an arbitrary $n$-qubit pure state among all
quantum states. We show that only $11$ Pauli measurements are needed to
determine an arbitrary two-qubit pure state compared to the full quantum state
tomography with $16$ measurements, and only $31$ Pauli measurements are needed
to determine an arbitrary three-qubit pure state compared to the full quantum
state tomography with $64$ measurements. We demonstrate that our protocol is
robust under depolarizing error with simulated random pure states. We
experimentally test the protocol on two- and three-qubit systems with nuclear
magnetic resonance techniques. We show that the pure state tomography protocol
saves us a number of measurements without considerable loss of fidelity. We
compare our protocol with same-size sets of randomly selected Pauli operators
and find that our selected set of Pauli measurements significantly outperforms
those random sampling sets. As a direct application, our scheme can also be
used to reduce the number of settings needed for pure-state tomography in
quantum optical systems.
