Non-invertible circuit complexity from fusion operations

TL;DR

This work extends quantum circuit complexity to systems with intrinsic non-invertible symmetries by modeling fusion as a completely positive, trace-preserving gate using a Stinespring dilation. Circuits then combine continuous unitary evolution within superselection sectors with discrete sector jumps described by a fusion graph, turning the sector-changing optimisation into a weighted shortest-path problem. The Ising fusion example illustrates genuine branching and path dependence, showing that non-invertible gates introduce a discrete component to complexity absent in purely unitary frameworks. An AdS$_3$-inspired interpretation links fusion-induced sector changes to geometry-changing boundary data, offering a kinematic picture where discrete energy jumps reorganize the bulk holonomy rather than continuously deforming a single geometry. The framework applies beyond rational CFTs to any quantum system with fusion-structured superselection sectors, suggesting a general role for non-invertible symmetries in the organisation of circuit complexity.

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.

Figures (2)

• Figure 1: A unitary gate acts invertibly within a fixed superselection sector. By contrast, fusion with a defect $b$ defines a quantum channel whose Kraus operators $K^{(a,b)}_{c,\mu}$ correspond to junctions (intertwiners) mapping an input sector $a$ to admissible output sectors $c$. Retaining the channel label $\mu$ yields a reversible embedding into a larger Hilbert space, while discarding it produces a trace-preserving but non-unitary map on the physical Hilbert space.
• Figure 2: Schematic fusion graph of superselection sectors. Vertices denote sectors and directed edges correspond to admissible fusion operations, weighted by gate costs $w_i$. Distinct paths from an initial sector $a_0$ to a target sector $a_\star$ represent different fusion circuits with the same endpoints but different total costs. Circuit optimisation reduces to a shortest-path problem on this graph.