Motivated by the limited qubit capacity of current quantum systems, we study the quantum sample complexity of $k$-qubit quantum operators, i.e., operations applicable on only $k$ out of $d$ qubits. The problem is studied according to the quantum probably approximately correct (QPAC) model abiding by quantum mechanical laws such as no-cloning, state collapse, and measurement incompatibility. With the delicacy of quantum samples and the richness of quantum operations, one expects a significantly larger quantum sample complexity. This paper proves the contrary. We show that the quantum sample complexity of $k$-qubit quantum operations is comparable to the classical sample complexity of their counterparts (juntas), at least when $\frac{k}{d}\ll 1$. This is surprising, especially since sample duplication is prohibited, and measurement incompatibility would lead to an exponentially larger sample complexity with standard methods. Our approach is based on the Pauli decomposition of quantum operators and a technique that we name Quantum Shadow Sampling (QSS) to reduce the sample complexity exponentially. The results are proved by developing (i) a connection between the learning loss and the Pauli decomposition; (ii) a scalable QSS circuit for estimating the Pauli coefficients; and (iii) a quantum algorithm for learning $k$-qubit operators with sample complexity $O(\frac{k4^k}{\epsilon^2}\log d)$.

翻译：受当前量子系统有限量子比特容量的启发，我们研究了$k$-量子比特量子算符的量子样本复杂度，即仅作用于$d$个量子比特中$k$个的操作。该问题根据遵守量子力学定律（如不可克隆性、态坍缩和测量不相容性）的量子概率近似正确（QPAC）模型进行研究。考虑到量子样本的精细性和量子操作的丰富性，人们预期量子样本复杂度会显著增大。本文证明了相反结论。我们表明$k$-量子比特量子操作的量子样本复杂度与其经典对应物（juntas）的经典样本复杂度相当，至少在$\frac{k}{d}\ll 1$时如此。这一结果令人惊讶，尤其是考虑到样本复制被禁止，且测量不相容性会通过标准方法导致指数级增大的样本复杂度。我们的方法基于量子算符的Pauli分解以及一种名为量子阴影采样（QSS）的技术，该技术可将样本复杂度指数级降低。通过以下方式证明结果：（i）建立学习损失与Pauli分解之间的联系；（ii）用于估计Pauli系数的可扩展QSS电路；（iii）一种学习$k$-量子比特算符的量子算法，其样本复杂度为$O(\frac{k4^k}{\epsilon^2}\log d)$。