Proper Learning of Shallow All-to-All Quantum Circuits

This work considers a variation on the problem of learning shallow quantum circuits. Given query access to the circuit, as well as knowledge of its gate layout, we consider the task of learning the specific gates used in the circuit, producing an operationally-equivalent circuit matching this structure. Building on recent work for learning Haar random brickwork circuits, we identify a meta-algorithmic framework for learning broad classes of circuits based on iterative local gate inversions at the front and back of the circuit. We apply these techniques to study random, all-to-all, two-local circuits, and provide analytical and numerical evidence that this ensemble undergoes a sharp learnability transition at depth $d^* \sim \log_2 n + \log_2\log_2 n$ in the large size limit, based on an analysis of lightcone growth. These results have implications for recently proposed quantum cryptographic schemes based on the difficulty of circuit learning, though there are important distinctions with respect to our setting that suggest avenues for future study.