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.

翻译：我们提出若干从量子统计查询（QSQ）中学习幺正算子的算法，这些查询基于其Choi-Jamiolkowski态。量子统计查询刻画了具有有限量子资源的学习者的能力，该学习者仅接收测量期望值的含噪估计作为输入。我们的方法依赖于一种新技术，该技术通过单个量子统计查询估计幺正在Pauli字符串子集上的傅里叶质量，推广了此前针对均匀量子样本的结果。利用这一洞见，我们证明量子Goldreich-Levin算法可通过量子统计查询实现，而该算法的先前版本需要查询幺正算子及其逆的预言机。此外，我们证明在模型中，常总影响度的$\mathcal{O}(\log n)$-juntas和量子布尔函数是可高效学习的，且常深度电路可通过量子统计查询实现样本高效学习。相比之下，这些任务的所有先前算法都需要直接访问Choi-Jamiolkowski态或查询幺正算子的预言机。同时，我们的上界表明，这些幺正算子类对本地随机化系综的作用是可高效学习的。我们还证明，尽管存在这些积极结果，与对Choi-Jamiolkowski态的可分离测量相比，量子统计查询在某些任务中会导致指数级更大的样本复杂度。特别地，我们展示了学习一类相位预言机幺正算子的指数下界，以及检验信道幺正性的双指数下界，这些结果将先前针对量子态的论证适配到我们的设定中。最后，我们提出平均情况替代模型的新定义，揭示了我们的结果在混合量子机器学习中的潜在应用。