# On construction of projection operators

## TL;DR

This paper presents a method for constructing projection operators onto eigen-subspaces of symmetry operators, crucial for maintaining physical symmetries in quantum approximations, with applications in electronic structure and quantum computing.

## Contribution

It introduces a systematic approach to build projection operators based on algebraic structures of symmetry operators, enhancing variational methods in quantum physics.

## Key findings

- Provides a two-step method for projection operator construction
- Utilizes algebraic structures like groups and Lie algebras
- Potential to improve variational approaches in quantum systems

## Abstract

The problem of construction of projection operators on eigen-subspaces of symmetry operators is considered. This problem arises in many approximate methods for solving time-independent and time-dependent quantum problems, and its solution ensures proper physical symmetries in development of approximate methods. The projector form is sought as a function of symmetry operators and their eigenvalues characterizing the eigen-subspace of interest. This form is obtained in two steps: 1) identification of algebraic structures within a set of symmetry operators (e.g. groups and Lie algebras), and 2) construction of the projection operators for individual symmetry operators. The first step is crucial for efficient projection operator construction because it allows for using information on irreducible representations of the present algebraic structure. The discussed approaches have promise to stimulate further developments of variational approaches for electronic structure of strongly correlated systems and in quantum computing.