  Quantum computing is a promising approach to harnessing strong correlation in
molecular systems; however, current devices only allow for hybrid
quantum-classical algorithms with a shallow circuit depth, such as the
variational quantum eigensolver (VQE). In this study, we report the importance
of the Hamiltonian symmetry in constructing VQE circuits adaptively. This
treatment often violates symmetry, thereby deteriorating the convergence of
fidelity to the exact solution, and ultimately resulting in deeper circuits. We
demonstrate that symmetry-projection can provide a simple yet effective
solution to this problem, by keeping the quantum state in the correct symmetry
space, to reduce the overall gate operations. The scheme also reveals the
significance of preserving symmetry in computing molecular properties, as
demonstrated in our illustrative calculations.
