Clifford Circuits can be Properly PAC Learned if and only if $\textsf{RP}=\textsf{NP}$

TL;DR

This paper proves that proper PAC learning of Clifford circuits is computationally hard unless certain complexity class equalities hold, linking quantum circuit learnability to fundamental complexity theory conjectures.

Contribution

It establishes the first hardness results for proper PAC learning of Clifford circuits, connecting quantum learning complexity to classical complexity class separations.

Findings

Proper learning of CNOT circuits is hard unless RP=NP.

Proper learning of Clifford circuits is hard unless RP=NP.

Efficient proper quantum learning of these circuits implies NP is contained in RQP.

Abstract

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