  We consider the learnability of the quantum neural network (QNN) built on the
variational hybrid quantum-classical scheme, which remains largely unknown due
to the non-convex optimization landscape, the measurement error, and the
unavoidable gate errors introduced by noisy intermediate-scale quantum (NISQ)
machines. Our contributions in this paper are multi-fold. First, we derive the
utility bounds of QNN towards empirical risk minimization, and show that large
gate noise, few quantum measurements, and deep circuit depth will lead to the
poor utility bounds. This result also applies to the variational quantum
circuits with gradient-based classical optimization, and can be of independent
interest. We then prove that QNN can be treated as a differentially private
(DP) model. Thirdly, we show that if a concept class can be efficiently learned
by QNN, then it can also be effectively learned by QNN even with gate noise.
This result implies the same learnability of QNN whether it is implemented on
noiseless or noisy quantum machines. We last exhibit that the quantum
statistical query (QSQ) model can be effectively simulated by noisy QNN. Since
the QSQ model can tackle certain tasks with runtime speedup, our result
suggests that the modified QNN implemented on NISQ devices will retain the
quantum advantage. Numerical simulations support the theoretical results.
