  Variational quantum algorithms (VQAs) have been proposed as one of the most
promising approaches to demonstrate quantum advantage on noisy
intermediate-scale quantum (NISQ) devices. However, it has been unclear whether
VQAs can maintain quantum advantage under the intrinsic noise of the NISQ
devices, which deteriorates the quantumness. Here we propose a measure, called
fermionic nonlinearity, to quantify the classical simulatability of quantum
circuits designed for simulating fermionic Hamiltonians. Specifically, we
construct a Monte Carlo type classical algorithm based on the classical
simulatability of fermionic linear optics, whose sampling overhead is
characterized by the fermionic nonlinearity. As a demonstration of these
techniques, we calculate the upper bound of the fermionic nonlinearity of a
rotation gate generated by four fermionic modes under the dephasing noise.
Moreover, we estimate the sampling costs of the unitary coupled cluster singles
and doubles quantum circuits for hydrogen chains subject to the dephasing
noise. We find that, depending on the error probability and atomic spacing,
there are regions where the fermionic nonlinearity becomes very small or unity,
and hence the circuits are classically simulatable. We believe that our method
and results help to design quantum circuits for fermionic systems with
potential quantum advantages.
