  Recent quantum technologies utilize complex multidimensional processes that
govern the dynamics of quantum systems. We develop an adaptive
diagonal-element-probing compression technique that feasibly characterizes any
unknown quantum processes using much fewer measurements compared to
conventional methods. This technique utilizes compressive projective
measurements that are generalizable to arbitrary number of subsystems. Both
numerical analysis and experimental results with unitary gates demonstrate low
measurement costs, of order $O(d^2)$ for $d$-dimensional systems, and
robustness against statistical noise. Our work potentially paves the way for a
reliable and highly compressive characterization of general quantum devices.
