Generalized Complexity Distances and Non-Invertible Symmetries

• URL: http://arxiv.org/abs/2604.14275v1
• Date: Wed, 15 Apr 2026 18:00:00 GMT
• Title: Generalized Complexity Distances and Non-Invertible Symmetries
• Abstract summary: Non-invertible symmetries of a quantum field theory (QFT) are a natural generalization of unitary symmetries.<n>We show that such symmetry operations on states define a collection of quantum gates for a parallel quantum computation scheme.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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