  Variational quantum algorithms (VQAs) offer the most promising path to
obtaining quantum advantages via noisy intermediate-scale quantum (NISQ)
processors. Such systems leverage classical optimization to tune the parameters
of a parameterized quantum circuit (PQC). The goal is minimizing a cost
function that depends on measurement outputs obtained from the PQC.
Optimization is typically implemented via stochastic gradient descent (SGD). On
NISQ computers, gate noise due to imperfections and decoherence affects the
stochastic gradient estimates by introducing a bias. Quantum error mitigation
(QEM) techniques can reduce the estimation bias without requiring any increase
in the number of qubits, but they in turn cause an increase in the variance of
the gradient estimates. This work studies the impact of quantum gate noise on
the convergence of SGD for the variational eigensolver (VQE), a fundamental
instance of VQAs. The main goal is ascertaining conditions under which QEM can
enhance the performance of SGD for VQEs. It is shown that quantum gate noise
induces a non-zero error-floor on the convergence error of SGD (evaluated with
respect to a reference noiseless PQC), which depends on the number of noisy
gates, the strength of the noise, as well as the eigenspectrum of the
observable being measured and minimized. In contrast, with QEM, any arbitrarily
small error can be obtained. Furthermore, for error levels attainable with or
without QEM, QEM can reduce the number of required iterations, but only as long
as the quantum noise level is sufficiently small, and a sufficiently large
number of measurements is allowed at each SGD iteration. Numerical examples for
a max-cut problem corroborate the main theoretical findings.
