Non-invertible Nielsen circuits and 3d Ising gravity

TL;DR

We extend Nielsen circuit complexity to include non-invertible gates arising from fusion with topological defects, promoting fusion data to completely positive, trace-preserving channels between superselection sectors within a unitary modular tensor category. This replaces the continuous geodesic optimization on a single group manifold with a discrete shortest-path problem on the fusion graph, and introduces three costs: intrinsic gate cost, energy-weighted cost, and a post-selection cost for isolating a definite fusion outcome. The framework is illustrated in rational CFTs, notably Ising and $\widehat{su(2)}_k$ WZW categories, and gains a bulk interpretation in AdS$_3$ gravity where fusion-induced sector changes correspond to shock-like defects altering boundary Virasoro data and bulk holonomies. This connects categorical fusion, quantum channel theory, and gravitational physics, offering a discrete, decision-based perspective on circuit complexity in theories with non-invertible symmetries and potential extensions to SymTFT and higher dimensions.

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.

Figures (8)

• Figure 1: Schematic action of a fusion gate on Virasoro representations. A gate labelled by $b$ acts on an initial conformal family $a$, mapping it to admissible output families $c$ through fusion channels labelled by $\mu\in V^c_{ab}$. Vertical columns represent full Virasoro modules (entire towers of descendants), not a level-by-level map. The channel label $\mu$ encodes fusion data and is retained as an auxiliary degree of freedom in the Stinespring construction.
• Figure 2: Schematic comparison between a unitary gate and a fusion-induced isometric gate. Fusion enlarges the Hilbert space by junction degrees of freedom, which become an ancilla in the Stinespring representation.
• Figure 3: The dark black line is the "evolution" of the state, while the gray-scale lines denote gates. Every non-invertible action of a gate $b$ on a state $|a\rangle$ determines generically multiple outcomes $c$ such that $N_{ab}^c\neq 0$. Through the associators, one can construct two path from an initial state $|a\rangle$ to a target state $|y\rangle$. Note that on the RHS of the bottom figure, one first fuses two gates before applying one branch to the state $|a\rangle$. Above, the mirror circuit process.
• Figure 4: Depiction of the intrinsic gate cost for a fusion process with three channels. Vectors on the unit sphere represent normalised weight vectors $(\sqrt{p_1},\sqrt{p_2},\sqrt{p_3})$. The black unit vector denotes the unbiased reference ray corresponding to the uniform distribution over channels, while the purple vector indicates a deterministic reference in which a single channel is selected. The green and orange unit vectors illustrate two example fusion outcomes with non-uniform channel weights. The intrinsic gate cost is indicated by the solid arcs.
• Figure 5: Left: The set of integrable $\widehat{su(2)}_k$ representations, labelled by spin $j = 0, \tfrac{1}{2}, \dots, \tfrac{k}{2}$. The finite interval reflects the integrability constraint; fusion gates act by inducing transitions between these labels, as illustrated on the right. Right: Two distinct competing length-two fusion paths from $j=1$, which sets at the boundary of the integrable alcove, back to $j=1$ under repeated fusion with $b=\tfrac{1}{2}$: the intermediate channel can be $j_1=\tfrac{1}{2}$ or $j_1=\tfrac{3}{2}$. Faint paths and dots indicate alternative fusion outcomes.