# Exact and approximate symmetry projectors for the electronic structure problem on a quantum computer

## TL;DR

This paper introduces exact and approximate symmetry projection methods for the VQE algorithm in quantum computing, improving accuracy and symmetry adherence in electronic structure calculations without increasing circuit depth.

## Contribution

It presents new symmetry projection operators for VQE, comparing their efficiency and accuracy benefits over traditional symmetry constraint methods.

## Key findings

- Projection operators enhance wavefunction accuracy.
- Projection methods reduce quantum circuit depth.
- Projection operators are feasible within VQE framework.

## Abstract

Solving the electronic structure problem on a universal-gate quantum computer within the variational quantum eigensolver (VQE) methodology requires constraining the search procedure to a subspace defined by relevant physical symmetries. Ignoring symmetries results in convergence to the lowest eigenstate of the Fock space for the second quantized electronic Hamiltonian. Moreover, this eigenstate can be symmetry broken due to limitations of the wavefunction ansatz. To address this VQE problem, we introduce and assess methods of exact and approximate projection operators to irreducible eigen-subspaces of available physical symmetries. Feasibility of symmetry projection operators in the VQE framework is discussed, and their efficiency is compared with symmetry constraint optimization procedures. Generally, projectors introduce higher numbers of terms for VQE measurement compared to the constraint approach. On the other hand, the projection formalism improves accuracy of the variational wavefunction ansatz without introducing additional unitary transformations, which is beneficial for reducing depths of quantum circuits.