Solving ground state energy of symmetric Hamiltonian by symmetric VQE

Abstract

In this paper, a symmetric Variational Quantum Eigensolver (VQE) algorithm is introduced to solve the ground state energy of symmetric Hamiltonian efficiently. As the symmetry of Hamiltonian is always broken by the typical VQE, hence the symmetric VQE proposed here is to increase the efficiency of the VQE when solving the symmetric Hamiltonian by preserving the symmetry. This scheme makes use of the fact that when the unitaries are commuted with the symmetry projection operator, the transformations by the unitaries are always in the symmetric subspace. This narrows down the searching space from the whole Hilbert space to the symmetric subspace. Since the searching is done within the symmetric subspace, the initial input state must also in the symmetric subspace. By using this scheme to solve the ground state energy of the symmetric ising model, the number of parameters needed for optimization is lesser when compared to a similar circuit structure which does not preserve the symmetry.