Non-invertible Nielsen circuits and 3d Ising gravity

TL;DR

This paper extends Nielsen's quantum circuit complexity framework to include non-invertible operations from topological defect fusion, revealing new optimization problems and connections to AdS3 gravity.

Contribution

It introduces a novel non-invertible circuit model based on fusion channels, expanding the scope of quantum complexity and linking it to 3D gravity phenomena.

Findings

Fusion operations as quantum channels between sectors

Optimization reduces to shortest-path on fusion graphs

Connections to shock-like defects in AdS3 gravity

Abstract

We extend Nielsen's formulation of quantum circuit complexity to include intrinsically non-invertible operations. Such gates arise from fusion with topological defect operators and remove a basic limitation of symmetry-based circuits: the inability to change superselection sectors, or in two-dimensional CFTs, conformal families. We realise fusion operations as completely positive, trace-preserving quantum channels acting between sectors, with consistency ensured by the fusion and associator data of an underlying unitary modular tensor category. In contrast to standard Nielsen circuits, non-invertible circuits lead to an optimisation problem that is no longer governed by geodesics on a continuous group manifold but instead reduces to a discrete shortest-path problem on the fusion graph of superselection sectors. We illustrate the framework in representative rational conformal field theories. Finally, we interpret fusion-induced transitions as discrete changes in boundary stress-tensor data, corresponding to shock-like defects in AdS$_3$ gravity.