  The resources required to characterise the dynamics of engineered quantum
systems-such as quantum computers and quantum sensors-grow exponentially with
system size. Here we adapt techniques from compressive sensing to exponentially
reduce the experimental configurations required for quantum process tomography.
Our method is applicable to dynamical processes that are known to be
nearly-sparse in a certain basis and it can be implemented using only
single-body preparations and measurements. We perform efficient, high-fidelity
estimation of process matrices on an experiment attempting to implement a
photonic two-qubit logic-gate. The data base is obtained under various
decoherence strengths. We find that our technique is both accurate and noise
robust, thus removing a key roadblock to the development and scaling of quantum
technologies.
