Non-invertible Nielsen Circuits Advance 3d Ising Gravity Understanding with Fusion Graphs

The challenge of quantifying complexity in physical systems receives fresh attention with a new investigation into non-invertible Nielsen circuits. Saskia Demulder of CUNEF Universidad, Madrid, and colleagues demonstrate an extension to Nielsen’s circuit complexity formulation, incorporating operations that were previously excluded. These intrinsically non-invertible gates, arising from fusion with topological defects, overcome a fundamental limitation of conventional circuits by allowing transitions between distinct superselection sectors. This research is significant because it reframes the optimisation problem from a continuous geodesic calculation to a discrete shortest-path problem on a fusion graph, offering a novel approach to understanding complexity in two-dimensional conformal field theories and potentially revealing insights into shock-like defects within AdS spacetime.

Two-dimensional conformal field theories (CFTs) and their conformal families are investigated. Fusion operations are realised as completely positive, trace-preserving quantum channels acting between superselection sectors, with consistency ensured by the fusion and associator data of an underlying unitary modular tensor category. This approach differs from standard Nielsen circuits, as non-invertible circuits result in an optimisation problem no longer governed by geodesics on a continuous group manifold. Instead, the problem reduces to a discrete shortest-path problem on the fusion graph of superselection sectors. The framework is illustrated using representative rational conformal field theories, and fusion-induced transitions are interpreted as discrete changes in boundary stress.

Complexity, Holography and Three-Dimensional Gravity

This section of the paper provides a broad and conceptually rich overview of the theoretical landscape at the intersection of quantum gravity, holography, and quantum information. A central theme is the holographic principle, particularly in the context of three-dimensional gravity, where gravitational dynamics in a bulk spacetime are related to quantum field theories living on a lower-dimensional boundary. Within this framework, the authors explore modern ideas connecting black hole geometry to quantum complexity, especially through conjectures such as the equivalence between holographic complexity and bulk gravitational action. These ideas aim to quantify how difficult it is to prepare certain quantum states and to relate that difficulty to geometric features of spacetime.

Topological quantum field theories play an important role in this discussion, serving as a bridge between gravity, symmetry, and quantum matter. Because TQFTs are insensitive to local geometric details, they provide a natural language for describing global and topological aspects of quantum systems, including boundary dynamics and exotic phases of matter. The paper also highlights the growing importance of non-invertible symmetries, which generalize conventional symmetry concepts and have emerged as a powerful framework for understanding new phases of quantum matter and the internal structure of TQFTs. These ideas are increasingly relevant in both high-energy and condensed matter physics.

Three-dimensional gravity is used as a particularly tractable setting in which to explore these concepts. In this reduced-dimensional framework, black hole solutions such as the BTZ black hole allow for explicit and controlled studies of quantum gravitational phenomena. Time-dependent geometries, including Vaidya spacetimes, are employed to model black hole formation and shock waves, providing insight into dynamical processes such as information scrambling and the butterfly effect. These phenomena are closely tied to chaos, quantum information spreading, and the growth of complexity in holographic systems.

The discussion also draws on a wide range of mathematical and theoretical tools, including Chern–Simons theory, asymptotic symmetries, central charges in conformal field theory, and quantum extremal surfaces used in holographic entropy calculations. Concepts from tensor categories, modular functors, and subfactor theory are invoked to describe the algebraic structure of TQFTs and the behavior of anyons with exotic exchange statistics. Together, these tools help formalize the deep connections between symmetry, topology, and quantum information.

Overall, this excerpt functions as a background and literature review section, situating the authors’ work within an active and rapidly developing research area. By synthesizing prior results, defining key terminology, and highlighting open questions, it motivates the subsequent analysis and prepares the reader for a more focused investigation—likely centered on the role of complexity, symmetry, and topology in three-dimensional quantum gravity and black hole physics.

Non-Invertible Gates and Conformal Family Transitions

Scientists have extended Nielsen’s formulation of circuit complexity to incorporate intrinsically non-invertible operations, addressing a fundamental limitation of prior symmetry-based circuits. These non-invertible gates, realised through fusion with topological defect operators, enable transitions between superselection sectors, specifically, conformal families in two-dimensional conformal field theories. The research team realised fusion operations as completely positive, trace-preserving channels acting between these sectors, ensuring consistency through the fusion and associator data of a unitary modular tensor category. This approach moves beyond geodesic optimisation on continuous group manifolds, instead framing the problem as a discrete shortest-path calculation on the fusion graph of superselection sectors.

Experiments revealed that the composition of non-invertible gates follows fusion rules, expressed as Dai Daj = M k N ak aiaj Dak, where ‘ak’ labels simple objects within the category and ‘N’ represents fusion multiplicities. Physically, these gates are interpreted as local insertions of topological defect junctions at fixed circuit time, effectively acting as genuine gate operations on the Hilbert space. This yields a well-defined extension of Nielsen circuits, allowing for discrete changes in conformal families within a continuous symmetry circuit and enabling the assignment of intrinsic costs to these transitions. The work focuses on rational conformal field theories, leveraging their finite and precisely known fusion data encoded within a unitary modular tensor category for full technical control.

Measurements confirm that fusion-induced transitions correspond to discrete changes in boundary stress-tensor data, manifesting as shock-like defects in Anti-de Sitter (AdS) space. Within the Chern-Simons formulation of AdS3 gravity, these transitions induce localised modifications of bulk holonomy data, altering the global holonomy class without introducing propagating gravitational degrees of freedom. The study interprets these changes as instantaneous alterations in boundary stress-tensor data, analogous to shock waves, and draws connections to recent explorations of time-dependent probes of Ising gravity, suggesting minimal models provide a controlled environment for investigating quantum aspects of AdS3 gravity. The breakthrough delivers a framework where fusion itself is a non-invertible circuit operation acting directly on the physical Hilbert space, differing from its role in topological quantum computation where it is typically used for state preparation or measurement. Categorical consistency ensures unambiguous composition and cost assignment for these quantum channels, even though they are intrinsically non-unitary, opening avenues for exploring quantum gravity beyond semiclassical limits and providing a novel approach to understanding the relationship between boundary conformal field theories and bulk gravitational configurations.

Non-Invertible Gates Simplify Circuit Complexity Problem

This work extends the established framework of Nielsen’s circuit complexity to incorporate intrinsically non-invertible operations, addressing a limitation of prior symmetry-based circuits which could not alter superselection sectors. Researchers achieved this by modelling fusion operations, arising from topological defects, as completely positive, trace-preserving channels operating between these sectors, ensuring consistency through the underlying mathematical structure of unitary modular tensor categories. The central contribution lies in demonstrating that optimising circuits with these non-invertible gates transforms the complexity problem from a continuous geodesic optimisation on a group manifold to a discrete shortest-path problem on the fusion graph of superselection sectors. This framework was successfully illustrated using rational conformal field theories, and the induced transitions were interpreted as discrete changes in boundary stress-tensor data, corresponding to shock-like defects within Anti-de Sitter space. The authors acknowledge a limitation in the current formulation regarding the finite dimensionality of the spaces involved, which requires further investigation. Future research could explore the implications of these findings for holographic complexity and the precise relationship between fusion-induced transitions and the geometry of shock-like defects in greater detail.