  The variational quantum eigensolver (VQE) is one of the most representative
quantum algorithms in the noisy intermediate-size quantum (NISQ) era, and is
generally speculated to deliver one of the first quantum advantages for the
ground-state simulations of some non-trivial Hamiltonians. However, short
quantum coherence time and limited availability of quantum hardware resources
in the NISQ hardware strongly restrain the capacity and expressiveness of VQEs.
In this Letter, we introduce the variational quantum-neural hybrid eigensolver
(VQNHE) in which the shallow-circuit quantum ansatz can be further enhanced by
classical post-processing with neural networks. We show that VQNHE consistently
and significantly outperforms VQE in simulating ground-state energies of
quantum spins and molecules given the same amount of quantum resources. More
importantly, we demonstrate that for arbitrary post-processing neural
functions, VQNHE only incurs an polynomial overhead of processing time and
represents the first scalable method to exponentially accelerate VQE with
non-unitary post-processing that can be efficiently implemented in the NISQ
era.
