Measuring global state properties with simple random single-qubit rotations

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 measurements of the system required to estimate these global quantities scales exponentially with system size. The use of random rotations can greatly reduce the basis of this exponent. We focus on experimentally relevant protocols for state tomography and state purity measurement in superconducting qubits, relying on simple X/Y rotations only. We show a framework for evaluating such protocols, and analytically calculate the required number of measurements, finding significant improvement in the scaling behavior.