Ungauging quantum error-correcting codes

TL;DR

This paper develops a systematic, chain-complex–based framework for gauging and ungauging Pauli CSS subsystem codes and demonstrates how these procedures map code subspaces into symmetry-constrained final spaces. Applying the framework to the 3D gauge color code, it identifies six decoupled copies of $\mathbb{Z}_2$ lattice gauge theory with $1$-form symmetries and shows that different stabilizer Hamiltonians realize distinct thermal SPT phases, including the Raussendorf–Bravyi–Harrington model at nonzero temperature. It also introduces fracton symmetry-protected topological phases by ungauging gapped domain walls in fractal codes and provides a general method to construct $D$-dimensional SPT phases from $(D+1)$-dimensional CSS stabilizer codes via domain walls and partial gauging. Collectively, the results connect fault-tolerant quantum computation, topological order, and finite-temperature symmetry-protected phases, and they offer a versatile route to generate and classify fracton and subsystem SPTs across dimensions.

Abstract

We develop the procedures of gauging and ungauging, reveal their operational meaning and propose their generalization in a systematic manner within the framework of quantum error-correcting codes. We demonstrate with an example of the subsystem Bacon-Shor code that the ungauging procedure can result in models with unusual symmetry operators constrained to live on lower-dimensional structures. We apply our formalism to the three-dimensional gauge color code (GCC) and show that its codeword space is equivalent to the Hilbert space of six copies of $\mathbb{Z}_2$ lattice gauge theory with $1$-form symmetries. We find that three different stabilizer Hamiltonians associated with the GCC correspond to distinct thermal symmetry-protected topological (SPT) phases in the presence of the stabilizer symmetries of the GCC. One of the considered Hamiltonians describes the Raussendorf-Bravyi-Harrington model, which is universal for measurement-based quantum computation at non-zero temperature. We also propose a general procedure of creating $D$-dimensional SPT Hamiltonians from $(D+1)$-dimensional CSS stabilizer Hamiltonians by exploiting a relation between gapped domain walls and transversal logical gates. As a result, we find an explicit two-dimensional realization of a non-trivial fracton SPT phase protected by fractal-like symmetries.

Figures (8)

• Figure 1: A chain complex associated with a 2D square lattice $\mathcal{L}$. The $\mathbb{F}_2$-vector spaces $C_2$, $C_1$, and $C_0$ are associated with faces, edges and vertices of $\mathcal{L}$. The boundary operator $\partial_2 : C_2 \rightarrow C_1$ is a linear map, which for each face returns the edges around that face. Similarly, $\partial_1: C_1 \rightarrow C_0$ is a linear map, which returns the vertices belonging to the given edge. We remark that binary vectors $c_i \in C_i$ (shaded in red) can be thought of as subsets of $i$-dimensional elements of the lattice $\mathcal{L}$.
• Figure 2: The ungauging map $\widetilde{\Gamma}$ and the gauging map $\Gamma$ are isomorphisms between symmetric subspaces $\mathcal{H}(\mathcal{S}_{\textrm{ini}}^Z)$ and $\mathcal{H}(\mathcal{S}_{\textrm{fin}}^X)$ of two Hilbert spaces $\mathcal{H}_{\textrm{ini}}$ and $\mathcal{H}_{\textrm{fin}}$. The subspaces $\mathcal{H}(\mathcal{S}_{\textrm{ini}}^Z)$ and $\mathcal{H}(\mathcal{S}_{\textrm{fin}}^X)$ are defined by the initial $Z$-type $\mathcal{S}^Z_{\textrm{ini}}$ and the emergent $X$-type $\mathcal{S}^X_{\textrm{fin}}$ symmetry groups. The ungauging map $\widetilde{\Gamma}$ transforms operators from the symmetric Pauli subgroup $\mathcal{P}(\mathcal{S}^Z_{\textrm{ini}})$ into operators in $\mathcal{P}(\mathcal{S}^Z_{\textrm{ini}})$; the gauging map $\Gamma$ can be viewed as an inverse of $\widetilde{\Gamma}$. Ungauging eliminates the initial $Z$-type symmetry group $\mathcal{S}^Z_{\textrm{ini}}$, whereas gauging eliminates the emergent $X$-type symmetry group $\mathcal{S}^X_{\textrm{fin}}$.
• Figure 3: The symmetry groups $\mathcal{S}_{\textrm{ini}}$ and $\mathcal{S}_{\textrm{fin}}$ of the initial and final system. The ungauging $\widetilde{\Gamma}$ and $\Gamma$ maps transform correspondingly the initial $Z$-type symmetries $\mathcal{S}^Z_{\textrm{ini}}$ and the emergent $X$-type symmetries $\mathcal{S}^X_{\textrm{fin}}$ into the identity operator. If there are some additional $X$-type symmetries $\mathcal{S}^X_{\textrm{ini}}$ in the initial system, then they will be preserved and mapped by $\widetilde{\Gamma}$ into the preserved $X$-type symmetries $\mathcal{S}^X_{\textrm{pre}}$ in the final system. Similarly, if we gauge the symmetry group $\mathcal{S}^X_{\textrm{fin}}$ of the final system, then $\Gamma$ would map $\mathcal{S}^X_{\textrm{pre}}$ into $\mathcal{S}^X_{\textrm{ini}}$ in the initial system.
• Figure 4: (a) The subsystem Bacon-Shor code with qubits placed on vertices of a square lattice on a torus. The $X$- and $Z$-type gauge generators are identified with horizontal $e_H$ (red) and vertical $e_V$ (blue) edges, respectively. We depict an $X$-type stabilizer (dashed red) and a representative of the bare logical $\overline Z$ operator (dashed blue). (b) The Xu-Moore model with qubits on horizontal edges. The Hamiltonian $H_{XM}$ of the model is a sum of single-qubit $X$ and four-qubit $Z$ (blue) operators. The symmetries of the model are $X$-type horizontal and vertical operators (dashed red).
• Figure 5: Examples of stars and links in 2D and 3D. In (a) and (b), we illustrate the $2$-star ${\mathrm{St}_{2}(v)}$ and the $3$-star ${\mathrm{St}_{3}(v)}$ of the vertex $v$ (red), which respectively are the sets of six triangular faces and eight tetrahedra (shaded in green) containing $v$. (c) The $1$-link ${\mathrm{Lk}_{1}(e)}$ of the edge $e$ (red) is the set of five edges (green), each of which belongs to the same tetrahedron as $e$ but does not overlap with $e$. (d) The $3$-star ${\mathrm{St}_{3}(e)}$ of the edge $e$ (red) is the set of five tetrahedra (shaded in green) containing $e$.