This work studies the learnability of unknown quantum circuits in the near term. We prove the natural robustness of quantum statistical queries for learning quantum processes and provide an efficient way to benchmark various classes of noise from statistics, which gives us a powerful framework for developing noise-tolerant algorithms. We adapt a learning algorithm for constant-depth quantum circuits to the quantum statistical query setting with a small overhead in the query complexity. We prove average-case lower bounds for learning random quantum circuits of logarithmic and higher depths within diamond distance with statistical queries. Additionally, we show the hardness of the quantum threshold search problem from quantum statistical queries and discuss its implications for the learnability of shallow quantum circuits. Finally, we prove that pseudorandom unitaries (PRUs) cannot be constructed using circuits of constant depth by constructing an efficient distinguisher and proving a new variation of the quantum no-free lunch theorem.

We propose several algorithms for learning unitary operators from quantum statistical queries (QSQs) with respect to their Choi-Jamiolkowski state. Quantum statistical queries capture the capabilities of a learner with limited quantum resources, which receives as input only noisy estimates of expected values of measurements. Our methods hinge on a novel technique for estimating the Fourier mass of a unitary on a subset of Pauli strings with a single quantum statistical query, generalizing a previous result for uniform quantum examples. Exploiting this insight, we show that the quantum Goldreich-Levin algorithm can be implemented with quantum statistical queries, whereas the prior version of the algorithm involves oracle access to the unitary and its inverse. Moreover, we prove that $\mathcal{O}(\log n)$-juntas and quantum Boolean functions with constant total influence are efficiently learnable in our model, and constant-depth circuits are learnable sample-efficiently with quantum statistical queries. On the other hand, all previous algorithms for these tasks require direct access to the Choi-Jamiolkowski state or oracle access to the unitary. In addition, our upper bounds imply that the actions of those classes of unitaries on locally scrambled ensembles can be efficiently learned. We also demonstrate that, despite these positive results, quantum statistical queries lead to an exponentially larger sample complexity for certain tasks, compared to separable measurements to the Choi-Jamiolkowski state. In particular, we show an exponential lower bound for learning a class of phase-oracle unitaries and a double exponential lower bound for testing the unitarity of channels, adapting to our setting previous arguments for quantum states. Finally, we propose a new definition of average-case surrogate models, showing a potential application of our results to hybrid quantum machine learning.

Quantum statistical queries provide a theoretical framework for investigating the computational power of a learner with limited quantum resources. This model is particularly relevant in the current context, where available quantum devices are subject to severe noise and have limited quantum memory. On the other hand, the framework of quantum differential privacy demonstrates that noise can, in some cases, benefit the computation, enhancing robustness and statistical security. In this work, we establish an equivalence between quantum statistical queries and quantum differential privacy in the local model, extending a celebrated classical result to the quantum setting. Furthermore, we derive strong data processing inequalities for the quantum relative entropy under local differential privacy and apply this result to the task of asymmetric hypothesis testing with restricted measurements. Finally, we consider the task of quantum multi-party computation under local differential privacy. As a proof of principle, we demonstrate that the parity function is efficiently learnable in this model, whereas the corresponding classical task requires exponentially many samples.