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arXiv:2601.09535

Non-invertible circuit complexity from fusion operations

Saskia Demulder

hep-thquant-phSubmitted Jan 14, 2026• Announced Jan 15, 2026• 0 citations
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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.

Comments
6 pages, 2 figures

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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 AdS3_33​ interpretation in which fusion-induced transitions correspond to geometry-changing boundary operations. A companion paper provides full derivations and extended examples.
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 AdS3_33​ gravity.
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.
A bstract 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.
In recent years, the traditional notion of symmetry in quantum theory was expanded to so-called generalised or categorical symmetries, which, unlike ordinary group symmetries, may be non-invertible. This appears to be at odds with Wigner's theorem, which requires quantum symmetries to be implemented by (anti)unitary -- and hence invertible -- operators in order to preserve probabilities. We resolve this puzzle for (higher) fusion category symmetries C\mathcal{C}C by proposing that, instead of acting by unitary operators on a fixed Hilbert space, symmetry defects in C\mathcal{C}C act as isometries between distinct Hilbert spaces constructed from twisted sectors. As a result, we find that non-invertible symmetries naturally act as trace-preserving quantum channels. Crucially, our construction relies on the symmetry category C\mathcal{C}C being unitary. We illustrate our proposal through several examples that include Tambara-Yamagami, Fibonacci, and Yang-Lee as well as higher categorical symmetries.
The notion of quantum symmetry has recently been extended to include reduced-dimensional transformations and algebraic structures beyond groups. Such generalized symmetries lead to exotic phases of matter and excitations that defy Landau's original paradigm. Here, we develop an algebraic framework for systematically deriving the fusion rules of topological defects in higher-dimensional lattice systems with non-invertible generalized symmetries, and focus on a (2+1)-dimensional quantum Ising plaquette model as a concrete illustration. We show that bond-algebraic automorphisms, when combined with the so-called half-gauging procedure, reveal the structure of the non-invertible duality symmetry operators, which can be explicitly represented as a sequential quantum circuit. The resulting duality defects are constrained by the model's rigid higher symmetries (lower-dimensional subsystem symmetries), leading to restricted mobility. We establish the fusion algebra of these defects. Finally, in constructing the non-invertible duality transformation, we explicitly verify that it acts as a partial isometry on the physical Hilbert space, thereby satisfying a recent generalization of Wigner's theorem applicable to non-invertible symmetries.
We propose an index of non-invertible symmetry operators in 1+1 dimensions and discuss its relation to the realizability of non-invertible symmetries on the tensor product of finite dimensional on-site Hilbert spaces on the lattice. Our index generalizes the Gross-Nesme-Vogts-Werner index of invertible symmetry operators represented by quantum cellular automata (QCAs). Assuming that all fusion channels of symmetry operators have the same index, we show that the fusion rules of finitely many symmetry operators on a tensor product Hilbert space can agree, up to QCAs, only with those of weakly integral fusion categories. We also discuss an attempt to establish an index theory for non-invertible symmetries within the framework of tensor networks. To this end, we first propose a general class of matrix product operators (MPOs) that describe non-invertible symmetries on a tensor product Hilbert space. These MPOs, which we refer to as topological injective MPOs, include all invertible symmetries, non-anomalous fusion category symmetries, and the Kramers-Wannier symmetries for finite abelian groups. For topological injective MPOs, we construct the defect Hilbert spaces and the corresponding sequential quantum circuit representations. We also show that all fusion channels of topological injective MPOs have the same index if there exist fusion and splitting tensors that satisfy appropriate conditions. The existence of such fusion and splitting tensors has not been proven in general, although we construct them explicitly for all examples of topological injective MPOs listed above.