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Quantum state tomography and other measures of the global properties of aquantum state are indispensable tools in understanding many body physicsthrough quantum simulators. Unfortunately, the number of experimentalmeasurements of the system required to estimate these global quantities scalesexponentially with system size. Here, we consider the use of random-axismeasurements for quantum state tomography and state purity estimation. Weperform a general analysis of the statistical deviation in such methods for anygiven algorithm. We then propose a simple protocol which relies on single-pulseX/Y rotations only. We find that it reduces the basis of the exponentialgrowth, 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.

Fourier-style Quantum State Tomography and Purity Measurement of a Multi-qubit System from Bloch Rotations