  The variational quantum eigensolver (VQE) is one of the most promising
algorithms to find eigenvalues and eigenvectors of a given Hamiltonian on noisy
intermediate-scale quantum (NISQ) devices. A particular application is to
obtain ground or excited states of molecules. The practical realization is
currently limited by the complexity of quantum circuits. Here we present a
novel approach to reduce quantum circuit complexity in VQE for electronic
structure calculations. Our algorithm, called ClusterVQE, splits the initial
qubit space into subspaces (qubit clusters) which are further distributed on
individual (shallower) quantum circuits. The clusters are obtained based on
quantum mutual information reflecting maximal entanglement between qubits,
whereas entanglement between different clusters is taken into account via a new
"dressed" Hamiltonian. ClusterVQE therefore allows exact simulation of the
problem by using fewer qubits and shallower circuit depth compared to standard
VQE at the cost of additional classical resources. In addition, a new gradient
measurement method without using an ancillary qubit is also developed in this
work. Proof-of-principle demonstrations are presented for several molecular
systems based on quantum simulators as well as an IBM quantum device with
accompanying error mitigation. The efficiency of the new algorithm is
comparable to or even improved over qubit-ADAPT-VQE and iterative Qubit Coupled
Cluster (iQCC), state-of-the-art circuit-efficient VQE methods to obtain
variational ground state energies of molecules on NISQ hardware. Above all, the
new ClusterVQE algorithm simultaneously reduces the number of qubits and
circuit depth, making it a potential leader for quantum chemistry simulations
on NISQ devices.
