  Variational hybrid quantum-classical algorithms (VHQCAs) are near-term
algorithms that leverage classical optimization to minimize a cost function,
which is efficiently evaluated on a quantum computer. Recently VHQCAs have been
proposed for quantum compiling, where a target unitary $U$ is compiled into a
short-depth gate sequence $V$. In this work, we report on a surprising form of
noise resilience for these algorithms. Namely, we find one often learns the
correct gate sequence $V$ (i.e., the correct variational parameters) despite
various sources of incoherent noise acting during the cost-evaluation circuit.
Our main results are rigorous theorems stating that the optimal variational
parameters are unaffected by a broad class of noise models, such as measurement
noise, gate noise, and Pauli channel noise. Furthermore, our numerical
implementations on IBM's noisy simulator demonstrate resilience when compiling
the quantum Fourier transform, Toffoli gate, and W-state preparation. Hence,
variational quantum compiling, due to its robustness, could be practically
useful for noisy intermediate-scale quantum devices. Finally, we speculate that
this noise resilience may be a general phenomenon that applies to other VHQCAs
such as the variational quantum eigensolver.
