Generalized Complexity Distances and Non-Invertible Symmetries

TL;DR

This paper explores non-invertible symmetries in quantum field theories, framing them as quantum gates, and introduces new measures of complexity and distance for these symmetries, with applications to specific QFTs.

Contribution

It generalizes the concept of symmetry distances and complexity measures to non-invertible symmetries, connecting them to quantum computation frameworks.

Findings

Non-invertible symmetries can be viewed as quantum gates.

Complexity measures extend to non-invertible symmetry structures.

Distances between symmetries in QFTs reveal high computational complexity.

Abstract

Non-invertible symmetries of a quantum field theory (QFT) are a natural generalization of unitary symmetries, but in which the product of operators does not satisfy a group multiplication law. We show that such symmetry operations on states define a collection of quantum gates for a parallel quantum computation scheme that includes post-selection / projection as a gate. Structures such as gate complexity and more geometric complexity measures generalize to this setting. We provide a class of distance / distinguishability measures that extend the standard notion of distance for Lie groups to both continuous and discrete non-invertible symmetries, as well as more general linear combinations of unitary quantum gates. We illustrate these considerations by computing the distance between non-invertible symmetries in some 4D and 2D QFTs. We find that the simple objects of a symmetry category can be highly complex computationally.