  We propose a scheme to restore spatial symmetry of Hamiltonian in the
variational-quantum-eigensolver (VQE) algorithm for which the quantum circuit
structures used usually break the Hamiltonian symmetry. The symmetry-adapted
VQE scheme introduced here simply applies the projection operator, which is
Hermitian but not unitary, to restore the spatial symmetry in a desired
irreducible representation of the spatial group. The entanglement of a quantum
state is still represented in a quantum circuit but the nonunitarity of the
projection operator is treated classically as postprocessing in the VQE
framework. By numerical simulations for a spin-$1/2$ Heisenberg model on a
one-dimensional ring, we demonstrate that the symmetry-adapted VQE scheme with
a shallower quantum circuit can achieve significant improvement in terms of the
fidelity of the ground state and has a great advantage in terms of the
ground-state energy with decent accuracy, as compared to the
non-symmetry-adapted VQE scheme. We also demonstrate that the present scheme
can approximate low-lying excited states that can be specified by symmetry
sectors, using the same circuit structure for the ground-state calculation.
