Non-invertible circuit complexity from fusion operations

TL;DR

This paper extends quantum circuit complexity to include non-invertible fusion operations, unifying continuous and discrete aspects and applying the framework to conformal field theories and holography.

Contribution

It introduces a novel approach to quantum circuit complexity that incorporates non-invertible fusion gates as quantum channels, bridging continuous geometry and discrete sector transitions.

Findings

Fusion operations are modelled as quantum channels.

Circuit complexity becomes a weighted shortest-path problem.

Application to rational conformal field theories and holography.

Abstract

Modern understanding of symmetry in quantum field theory includes both invertible and non-invertible operations. Motivated by this, we extend Nielsen's geometric approach to quantum circuit complexity to incorporate non-invertible gates. These arise naturally from fusion of topological defects and allow transitions between superselection sectors. We realise fusion operations as completely positive, trace-preserving quantum channels. Including such gates makes the sector-changing optimisation problem discrete: it reduces to a weighted shortest-path problem on the fusion graph. Circuit complexity therefore combines continuous geometry within sectors with discrete sector jumps. We illustrate the framework in rational conformal field theories and briefly comment on an AdS$_3$ interpretation in which fusion-induced transitions correspond to geometry-changing boundary operations. A companion paper provides full derivations and extended examples.