Anomaly inflow for CSS and fractonic lattice models and dualities via cluster state measurement

TL;DR

This work develops a unified framework linking Calderbank-Shor-Steane (CSS) codes to foliated cluster states, demonstrating that the foliated states realize symmetry-protected topological order protected by generalized global symmetries on cycles of the foliated CSS chain complex. It proves an anomaly inflow between generic CSS codes and their foliated cluster-state boundaries, formulates a bulk–boundary correspondence via defect insertions, and derives a generalized Kramers-Wannier-Wegner duality for associated statistical models, including strange correlators for fracton systems. The paper applies the construction to toric code, RBH, X-cube, and checkerboard fracton models, providing explicit KW dualities, non-invertible symmetry structures, and self-dual subsystem models, all within a BF/SymTFT perspective. These results yield a powerful, geometry-heavy toolkit for analyzing dualities, anomalies, and boundary phenomena across CSS/fracton platforms, with potential applications to measurement-based quantum computation and beyond.

Abstract

Calderbank-Shor-Steane (CSS) codes are a class of quantum error correction codes that contains the toric code and fracton models. A procedure called foliation defines a cluster state for a given CSS code. We use the CSS chain complex and its tensor product with other chain complexes to describe the topological structure in the foliated cluster state, and argue that it has a symmetry-protected topological order protected by generalized global symmetries supported on cycles in the foliated CSS chain complex. We demonstrate the so-called anomaly inflow between CSS codes and corresponding foliated cluster states by explicitly showing the equality of the gauge transformations of the bulk and boundary partition functions defined as functionals of defect world-volumes. We show that the bulk and boundary defects are related via measurement of the bulk system. Further, we provide a procedure to obtain statistical models associated with general CSS codes via the foliated cluster state, and derive a generalization of the Kramers-Wannier-Wegner duality for such statistical models with insertion of twist defects. We also study the measurement-assisted gauging method with cluster-state entanglers for CSS/fracton models based on recent proposals in the literature, and demonstrate a non-invertible fusion of duality operators. Using the cluster-state entanglers, we construct the so-called strange correlator for general CSS/fracton models. Finally, we introduce a new family of subsystem-symmetric quantum models each of which is self-dual under the generalized Kramers-Wannier-Wegner duality transformation, which becomes a non-invertible symmetry.

Figures (13)

• Figure 1: The foliated cluster state with edge modes satisfies the relations in \ref{['eq:XtoXX-inflow']} and \ref{['eq:XtoZZ-inflow']}. The logical operator in \ref{['eq:XtoXX-inflow']} is supported on $z^{*(0)}_{\rm q}$ and $z^{*(L_w)}_{\rm q}$, which are elements in $\text{Ker}\,(\delta^*_Z)$. On the other hand, the logical operator in \ref{['eq:XtoZZ-inflow']} is supported on $z^{(0)}_{\rm q}$ and $z^{(L_w)}_{\rm q}$, which are elements in $\text{Ker}\,(\delta_X)$.
• Figure 2: The unit cell of the cubic lattice on which the bulk cluster state for the $(2+1)$d plaquette Ising model is defined. The horizontal layer for $w=j$ ($j\in\mathbb{Z}$) corresponds to the face stabilizers $A_f= \prod_{v\subset f} Z_v$ and contains qubits corresponding to code qubits (on vertices) and $Z$-stabilizers (on horizontal faces). The horizontal layer for $w=j+1/2$ contains only qubits corresponding to code qubits (on vertical edges).
• Figure 3: An example of excitations in $H^{\text{2d-qPIM}}_{\mathcal{C}}$ described by a dual cycle $\bm z^{*}_{Q_1}$.
• Figure 4: Symmetries in the cluster state $H^{\text{2d-qPIM}}_{\mathcal{C}}$. (Left) An example of symmetry generators supported on $\bm{z}_{Q_1}$. (Right) An example of symmetry generators supported on $\bm{z}^*_{Q_2}$.
• Figure 5: The X-cube model.