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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2601.09534 |
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| _version_ | 1866918289223974912 |
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| author | Demulder, Saskia |
| author_facet | Demulder, Saskia |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09534 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-invertible Nielsen circuits and 3d Ising gravity Demulder, Saskia High Energy Physics - Theory 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. |
| title | Non-invertible Nielsen circuits and 3d Ising gravity |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2601.09534 |