  The variational quantum-classical algorithms are the most promising approach
for achieving quantum advantage on near-term quantum simulators. Among these
methods, the variational quantum eigensolver has attracted a lot of attention
in recent years. While it is very effective for simulating the ground state of
many-body systems, its generalization to excited states becomes very resource
demanding. Here, we show that this issue can significantly be improved by
exploiting the symmetries of the Hamiltonian. The improvement is even more
effective for higher energy eigenstates. We introduce two methods for
incorporating the symmetries. In the first approach, called hardware symmetry
preserving, all the symmetries are included in the design of the circuit. In
the second approach, the cost function is updated to include the symmetries.
The hardware symmetry preserving approach indeed outperforms the second
approach. However, integrating all symmetries in the design of the circuit
could be extremely challenging. Therefore, we introduce hybrid symmetry
preserving method in which symmetries are divided between the circuit and the
classical cost function. This allows to harness the advantage of symmetries
while preventing sophisticated circuit design.
