  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 $\textsf{NP} \subseteq \textsf{RQP}$.
