Symmetry-adapted variational quantum eigensolver

TL;DR

This paper introduces a symmetry-adapted VQE method that applies a classical projection to restore spatial symmetry in quantum circuits, improving accuracy and efficiency in finding ground and excited states.

Contribution

It proposes a novel symmetry restoration technique in VQE using a classical projection operator, enhancing ground state fidelity and energy accuracy with shallower circuits.

Findings

Significant improvement in ground state fidelity.

Enhanced ground-state energy accuracy.

Ability to approximate excited states in specific symmetry sectors.

Abstract

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.