  Applying low-depth quantum neural networks (QNNs), variational quantum
algorithms (VQAs) are both promising and challenging in the noisy
intermediate-scale quantum (NISQ) era: Despite its remarkable progress,
criticisms on the efficiency and feasibility issues never stopped. However,
whether VQAs can demonstrate quantum advantages is still undetermined till now,
which will be investigated in this paper. First, we will prove that there
exists a dependency between the parameter number and the gradient-evaluation
cost when training QNNs. Noticing there is no such direct dependency when
training classical neural networks with the backpropagation algorithm, we argue
that such a dependency limits the scalability of VQAs. Second, we estimate the
time for running VQAs in ideal cases, i.e., without considering realistic
limitations like noise and reachability. We will show that the ideal time cost
easily reaches the order of a 1-year wall time. Third, by comparing with the
time cost using classical simulation of quantum circuits, we will show that
VQAs can only outperform the classical simulation case when the time cost
reaches the scaling of $10^0$-$10^2$ years. Finally, based on the above
results, we argue that it would be difficult for VQAs to outperform classical
cases in view of time scaling, and therefore, demonstrate quantum advantages,
with the current workflow. Since VQAs as well as quantum computing are
developing rapidly, this work does not aim to deny the potential of VQAs. The
analysis in this paper provides directions for optimizing VQAs, and in the long
run, seeking more natural hybrid quantum-classical algorithms would be
meaningful.
