  Variational quantum algorithms (VQAs) are expected to become a practical
application of near-term noisy quantum computers. Although the effect of the
noise crucially determines whether a VQA works or not, the heuristic nature of
VQAs makes it difficult to establish analytic theories. Analytic estimations of
the impact of the noise are urgent for searching for quantum advantages, as
numerical simulations of noisy quantum computers on classical computers are
heavy and quite limited to small scale problems. In this paper, we establish
analytic estimations of the error in the cost function of VQAs due to the
noise. The estimations are applicable to any typical VQAs under the Gaussian
noise, which is equivalent to a class of stochastic noise models. Notably, the
depolarizing noise is included in this model. As a result, we obtain
estimations of the noise level to guarantee a required precision. Our formulae
show how the Hessian of the cost function, the spectrum of the target operator,
and the geometry of the ansatz affect the sensitivity to the noise. This
insight implies trade-off relations between the trainability and the noise
resilience of the cost function. We also obtain rough estimations which can be
easily calculated without detailed information of the cost function. As a
highlight of the applications of the formula, we propose a quantum error
mitigation method which is different from the extrapolation and the
probabilistic error cancellation.
