Clifford Circuits can be Properly PAC Learned if and only if $\textsf{RP}=\textsf{NP}$
This addresses a fundamental computational efficiency question in quantum machine learning, with implications for learning quantum circuits, but it is incremental as it builds on prior work on PAC learning quantum circuits.
The paper tackles the problem of efficiently predicting measurement probabilities for quantum circuits, specifically showing that proper PAC learning of CNOT and Clifford circuits is computationally hard for classical learners unless RP = NP, and similarly for quantum learners unless NP ⊆ RQP.
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 $\textsf{NP} \subseteq \textsf{RQP}$.