Quantum Algorithms for Realizing Symmetric, Asymmetric, and Antisymmetric Projectors

TL;DR

This paper introduces quantum algorithms for projecting onto symmetric, asymmetric, and antisymmetric subspaces, enabling efficient symmetry testing and state analysis in quantum systems, with practical demonstrations on IBM Quantum hardware.

Contribution

It presents novel quantum algorithms for symmetry projections, including systematic combinations for multiple projections, and demonstrates their application in quantum state analysis.

Findings

Successful implementation of symmetry projections on IBM Quantum systems.

Effective testing of Werner-state symmetry and estimation of Schmidt ranks.

Algorithms improve symmetry detection efficiency in quantum computing.

Abstract

In quantum computing, knowing the symmetries a given system or state obeys or disobeys is often useful. For example, Hamiltonian symmetries may limit allowed state transitions or simplify learning parameters in machine learning applications, and certain asymmetric quantum states are known to be resourceful in various applications. Symmetry testing algorithms provide a means to identify and quantify these properties with respect to a representation of a group. In this paper, we present a collection of quantum algorithms that realize projections onto the symmetric subspace, as well as the asymmetric subspace, of quantum systems. We describe how this can be modified to realize an antisymmetric projection as well, and we show how projectors can be combined in a systematic way to effectively measure various projections in a single quantum circuit. Using these constructions, we demonstrate applications such as testing for Werner-state symmetry and estimating Schmidt ranks of bipartite states, supported by experimental data from IBM Quantum systems. This work underscores the pivotal role of symmetry in simplifying quantum calculations and advancing quantum information tasks.