Given a dataset of input states, measurements, and probabilities, is it possible to efficiently predict the measurement probabilities associated with a quantum circuit? Recent work of Caro and Datta (2020) studied the problem of PAC learning quantum circuits in an information theoretic sense, leaving open questions of computational efficiency. In particular, one candidate class of circuits for which an efficient learner might have been possible was that of Clifford circuits, since the corresponding set of states generated by such circuits, called stabilizer states, are known to be efficiently PAC learnable (Rocchetto 2018). Here we provide a negative result, showing that proper learning of CNOT circuits is hard for classical learners unless $\textsf{RP} = \textsf{NP}$. As the classical analogue and subset of Clifford circuits, this naturally leads to a hardness result for Clifford circuits as well. Additionally, we show that if $\textsf{RP} = \textsf{NP}$ then there would exist efficient proper learning algorithms for CNOT and Clifford circuits. By similar arguments, we also find that an efficient proper quantum learner for such circuits exists if and only if $\textsf{NP} \subseteq \textsf{RQP}$.

翻译：给定一个由输入态、测量和概率组成的数据集，能否高效预测与量子电路相关的测量概率？Caro与Datta（2020）近期工作从信息论角度研究了PAC学习量子电路的问题，但计算效率问题仍未解决。特别地，一类可能实现高效学习的候选电路是Clifford电路——因其产生的对应量子态集合（称为稳定子态）已被证明可高效PAC学习（Rocchetto 2018）。本文给出否定结果：除非$\textsf{RP} = \textsf{NP}$，否则经典学习器无法正确学习CNOT电路。由于CNOT电路是Clifford电路的经典类比与子集，该结论自然推导出Clifford电路的学习困难性。此外，我们证明若$\textsf{RP} = \textsf{NP}$成立，则存在针对CNOT与Clifford电路的高效正确学习算法。通过类似论证，我们还发现此类电路存在高效量子正确学习器的充要条件是$\textsf{NP} \subseteq \textsf{RQP}$。