Unveiling dynamical quantum error correcting codes via non-invertible symmetries

TL;DR

This work addresses how to understand dynamical stabilizer codes (DSCs) through a topological quantum field theory lens by mapping qudit Pauli measurements to non-invertible 2-form symmetries in a $4+1$-dimensional TQFT. The main approach uses a correspondence between measurement-induced symmetry actions and endable surface operators, with detectors identified as endable surfaces and logical and detectable errors linked to non-endable surfaces that braid nontrivially with lines, thereby naturally reproducing spacetime stabilizer codes. Key contributions include a detailed DSC update rule framework, the construction of condensation operators $ ext{W}_B$, and explicit connections to the spacetime stabilizer code via cumulants/back-cumulants, all within a unified 2-form gauge theory. The results provide a principled, topological viewpoint on DSCs and suggest extensions to higher-form and non-invertible symmetry theories with potential implications for fault-tolerant gates and new code families.

Abstract

Dynamical stabilizer codes (DSCs) have recently emerged as a powerful generalization of static stabilizer codes for quantum error correction, replacing a fixed stabilizer group with a sequence of non-commuting measurements. This dynamical structure unlocks new possibilities for fault tolerance but also introduces new challenges, as errors must now be tracked across both space and time. In this work, we provide a physical and topological understanding of DSCs by establishing a correspondence between qudit Pauli measurements and non-invertible symmetries in 4+1-dimensional 2-form gauge theories. Sequences of measurements in a DSC are mapped to a fusion of the operators implementing these non-invertible symmetries. We show that the error detectors of a DSC correspond to endable surface operators in the gauge theory, whose endpoints define line operators, and that detectable errors are precisely those surface operators that braid non-trivially with these lines. Finally, we demonstrate how this framework naturally recovers the spacetime stabilizer code associated with a DSC.

Figures (8)

• Figure 1: Fusing the interface $\mathcal{I}_B$ with its dual $\mathcal{I}_B^{\dagger}$ gives a 4-dimensional operator $\mathcal{W}_B$ in the TQFT $\mathcal{T}$.
• Figure 2: Action of the 4-dimensional operator $\mathcal{W}_{\phi(\mathcal{S})}$ on 2-dimensional surface operator $U$. Note that we have drawn the diagram as a 2d operator acting on a 1d operator for clarity.
• Figure 3: Consider surface operators $U_1$ and $U_2$ which can from junction with the trivial surface operator $\mathds{1}$ as in this figure. On fusing the three operators $\mathcal{W}_{\phi(\mathcal{S})}$, we get a line operator $L_{U_1,U_2}$ on $\mathcal{W}_3$.
• Figure 4: A surface operator on $\mathcal{W}_3$ can be obtained from a non-unqiue configuration of surface operators $V_1$ and $V_2$ of the TQFT $\mathcal{T}$.
• Figure 5: A surface operators in $\mathcal{W}_3$ which braids non-trivially with $L_{U_1,U_2}$ for some $U_1,U_2$ is a detectable error.