  In this work, we give a hybrid quantum-classical algorithm for solving
electronic structure problems of molecules using only linear quantum optical
systems. The variational ansatz we proposed is a hybrid of non-interacting
Boson dynamics and classical computational chemistry methods, specifically, the
Hartree-Fock method and the Configuration Interaction method. The Boson part is
built by a linear optical interferometer which is easier to realize compared
with the well-known Unitary Coupled Cluster (UCC) ansatz composed of quantum
gates in conventional VQE and the classical part is merely classical processing
acting on the Hamiltonian. We called such ansatzes Boson Sampling-Classic
(BS-C). The appearance of permanents in the Boson part has its physical
intuition to provide different kinds of resources from commonly used single-,
double-, and higher-excitations in classical methods and the UCC ansatz to
exploring chemical quantum states. Such resources can help enhance the accuracy
of methods used in the classical parts. We give a scalable hybrid homodyne and
photon number measurement procedure for evaluating the energy value which has
intrinsic abilities to mitigate photon loss errors and discuss the extra
measurement cost induced by the no Pauli exclusion principle for Bosons with
its solutions. To demonstrate our proposal, we run numerical experiments on
several molecules and obtain their potential energy curves reaching chemical
accuracy.
