Variational quantum circuits to prepare low energy symmetry states

We explore how to build quantum circuits that compute the lowest energy state corresponding to a given Hamiltonian within a Symmetry subspace by explicitly encoding it into the circuit. We create an explicit unitary and a variationally trained unitary that maps any vector output by ansatz A( α) from a defined subspace to a vector in the symmetry space. The parameters are trained varitionally to minimize the energy thus keeping the output within the labelled symmetry value. The method was tested for a spin XXZ hamiltonian using rotation and reflection symmetry and H2 hamiltonian within S_z = 0 subspace using S^2 symmetry. We have found the variationally trained unitary surprisingly giving very good results with very low depth circuits and can thus be used to prepare symmetry states within near term quantum computers.