  Quantum Computing is believed to be the ultimate solution for quantum
chemistry problems. Before the advent of large-scale, fully fault-tolerant
quantum computers, the variational quantum eigensolver~(VQE) is a promising
heuristic quantum algorithm to solve real world quantum chemistry problems on
near-term noisy quantum computers. Here we propose a highly parallelizable
classical simulator for VQE based on the matrix product state representation of
quantum state, which significantly extend the simulation range of the existing
simulators. Our simulator seamlessly integrates the quantum circuit evolution
into the classical auto-differentiation framework, thus the gradients could be
computed efficiently similar to the classical deep neural network, with a
scaling that is independent of the number of variational parameters. As
applications, we use our simulator to study commonly used small molecules such
as HF, HCl, LiH and H$_2$O, as well as larger molecules CO$_2$, BeH$_2$ and
H$_4$ with up to $40$ qubits. The favorable scaling of our simulator against
the number of qubits and the number of parameters could make it an ideal
testing ground for near-term quantum algorithms and a perfect benchmarking
baseline for oncoming large scale VQE experiments on noisy quantum computers.
