  Quantum state tomography and other measures of the global properties of a
quantum state are indispensable tools in understanding many body physics
through quantum simulators. Unfortunately, the number of experimental
measurements of the system required to estimate these global quantities scales
exponentially with system size. Here, we consider the use of random-axis
measurements for quantum state tomography and state purity estimation. We
perform a general analysis of the statistical deviation in such methods for any
given algorithm. We then propose a simple protocol which relies on single-pulse
X/Y rotations only. We find that it reduces the basis of the exponential
growth, calculating the statistical variance to scale as
$\sum_{a,b}\left\lvert\Delta\rho_{ab}\right\rvert^{2}\sim 5^{N}/N_{\rm tot}$
for full tomography, and $\left(\Delta\mu\right)^{2}\sim 7^{N}/N_{\rm tot}^{2}$
for purity estimation, for $N$ qubits and $N_{\rm tot}$ measurements performed.
