  We consider the problem of learning stabilizer states with noise in the
Probably Approximately Correct (PAC) framework of Aaronson (2007) for learning
quantum states. In the noiseless setting, an algorithm for this problem was
recently given by Rocchetto (2018), but the noisy case was left open. Motivated
by approaches to noise tolerance from classical learning theory, we introduce
the Statistical Query (SQ) model for PAC-learning quantum states, and prove
that algorithms in this model are indeed resilient to common forms of noise,
including classification and depolarizing noise. We prove an exponential lower
bound on learning stabilizer states in the SQ model. Even outside the SQ model,
we prove that learning stabilizer states with noise is in general as hard as
Learning Parity with Noise (LPN) using classical examples. Our results position
the problem of learning stabilizer states as a natural quantum analogue of the
classical problem of learning parities: easy in the noiseless setting, but
seemingly intractable even with simple forms of noise.
