Symmetry breaking/symmetry preserving circuits and symmetry restoration on quantum computers: A quantum many-body perspective

TL;DR

This paper reviews symmetry concepts in quantum many-body systems on quantum computers, emphasizing symmetry preservation, breaking, and restoration techniques, and introduces new methods for effective symmetry restoration using quantum algorithms.

Contribution

It provides a comprehensive review of symmetry handling in quantum computing and proposes novel symmetry restoration methods based on the linear combination of unitaries approach.

Findings

Symmetry encoding reduces quantum register size.

Symmetry-breaking states help incorporate internal correlations.

New symmetry restoration methods using state purification are proposed.

Abstract

We discuss here some aspects related to the symmetries of a quantum many-body problem when trying to treat it on a quantum computer. Several features related to symmetry conservation, symmetry breaking, and possible symmetry restoration are reviewed. After briefly discussing some of the standard symmetries relevant for many-particle systems, we discuss the advantage of encoding some symmetries directly in quantum ans\"atze, especially to reduce the quantum register size. It is, however, well-known that the use of symmetry-breaking states can also be a unique way to incorporate specific internal correlations when a spontaneous symmetry breaking occurs. These aspects are discussed in the quantum computing context. Ultimately, an accurate description of quantum systems can be achieved only when the initially broken symmetries are properly restored. We review several methods explored previously to perform symmetry restoration on a quantum computer, for instance, the ones based on symmetry filtering by quantum phase estimation and by an iterative independent set of Hadamard tests. We propose novel methods that pave the new directions to perform symmetry restoration, like those based on the purification of the state employing the linear combination of unitaries (LCU) approach.