  Learning complex quantum processes is a central challenge in many areas of
quantum computing and quantum machine learning, with applications in quantum
benchmarking, cryptanalysis, and variational quantum algorithms. This paper
introduces the first learning framework for studying quantum process learning
within the Quantum Statistical Query (QSQ) model, providing the first formal
definition of statistical queries to quantum processes (QPSQs). The framework
allows us to propose an efficient QPSQ learner for arbitrary quantum processes
accompanied by a provable performance guarantee. We also provide numerical
simulations to demonstrate the efficacy of this algorithm. In our new
framework, we prove exponential query complexity lower bounds for learning
unitary 2-designs, and a doubly exponential lower bound for learning
haar-random unitaries. The practical relevance of this framework is exemplified
through application in cryptography, highlighting vulnerabilities of a large
class of Classical-Readout Quantum Physical Unclonable Functions (CR-QPUFs),
addressing an important open question in the field of quantum hardware
security. This work marks a significant step towards understanding the
learnability of quantum processes and shedding light on their security
implications.
