Lifting a CSS code via its handlebody realization

TL;DR

The paper connects topological code lifting via Freedman–Hastings handlebody realizations with the Tanner-cone complex approach, showing that manifold-based lifts and cone-complex lifts are essentially equivalent in many regimes. It then introduces a cellular realization framework and defines cellular-lifts through covers of finite CW complexes, clarifying the role of the $ ext{Z}$-lift and the conditions under which Tanner-lift and cellular-lift coincide. The authors establish a classification for lifts, particularly for hypergraph-product codes, by tying lift data to subgroups of fundamental groups (e.g., $ ext{π}_1(T_1) imes ext{π}_1(T_2)$) and discussing when such lifts preserve sparsity and support. The work yields a practical pathway to construct and classify lifted quantum LDPC codes from HPCs, extending the toolbox for realizing LDPC codes with favorable distance properties in a topological setting.

Abstract

We present a topological approach to lifting a quantum CSS code. In previous work, we proposed lifting a CSS code by constructing covering spaces over its 2D simplicial complex representation, known as the Tanner cone-complex. This idea was inspired by the work of Freedman and Hastings, which associates CSS codes with handlebodies. In this paper, we show how the handlebody realization of a code can also be used to perform code lifting, and we provide a more detailed discussion of why this is essentially equivalent to the Tanner cone-complex approach. As an application, we classify lifts of hypergraph-product codes via their handlebody realization.

Figures (6)

• Figure 1: A dimensionally reduced representation of a dressed 3-handle as a product bundle of a 3-sphere and an 8-disk. Sections of this bundle, represented as dashed spheres, are where the 4-dressed-handles can be attached (along their punctures).
• Figure 2: (a) Handle structure of a 4-dressed-handle. 1-handles are represented as segments, and 2-handles as disks. This corresponds to the cell structure when retracting handles to their core. (b) Modified cell structure discussed in Section \ref{['section Cellular realization of a code']}, with the addition of a 0-cell and a 1-cell. (c) Homeomorphic transformation of the dressed-handle, resulting in (d), a graph representation of the dressed-handle, with a blue vertex representing the 4-cell and half-edges corresponding to the punctures.
• Figure 3: (a) Example of a handle structure of a 5-dressed-handle, where its components are represented in green, and the 5-handle is represented by a 3-disk. The blue region is a boundary component of the manifold $M_{QX}$. It corresponds to a the boundary of a thickened graph $G_z$, with 2 connected components. The construction of the 5-dressed-handle consists in joining them with a 1-handle, and triviliazing the fundamental group by adding 2-handles. The 5-handle is attached at the end along the resulting boundary. (b) Dimensionally reduced representation of the resulting manifold, where the 1-handle is represented by a thick green line, and the 5-handle by a 2-disk.
• Figure 4: (a) Tanner graph of the code considered in Example \ref{['example 5-handle to MQX']}, with qubits shown in blue and $X$-checks in red. (b) Signed multigraph $\widetilde{T}$ of the code $C_X$, which also corresponds to the induced subgraph $\widetilde{T}_z$ in this example. Positively signed edges are depicted as thick lines, and negatively signed edges as thin lines. (c) The corresponding $\widetilde{T}_z^{\text{mult}}$, where each qubit appears in a number of copies determined by the corresponding entry of $H_Z$. (d) Pairing of positively and negatively signed edges at the $X$-vertices, resulting in the graph $G_z$ with Betti number $b_1 = 3$. This graph represents the manifold $Y_z$. (e) Manifold $Y_z = \#_3 (S^3 \times S^1)$ corresponding to the attaching region of the 5-dressed-handle $h_z$. The four blue components correspond to the qubit copies in $G_z$ (punctured spheres), while the six red spheres represent the $X$-vertices (3-handles).
• Figure 5: Manifold $M_{QX}$ in Example \ref{['example 5-handle to MQX']}, where the two 3-dressed-handles ($X$-checks) are shown in red, and the two 4-dressed-handles (qubits) in blue. This representation should be interpreted as a $4$-manifold crossed with a $7$-disk. Copies of manifolds of the form $Y_z$ are embedded within it, here represented by the graph $G_z$ (see Figure \ref{['fig:Tanner graphs']}). This embedded manifold serves as a valid attaching region for the 5-dressed-handle $h_z$.

Theorems & Definitions (33)

• Definition 2.1
• Proposition 2.2
• Definition 2.3: Tanner-lift of a CSS code, GuemardLiftIEEE
• Definition 2.4: cellular-lift
• Definition 2.5
• Definition 2.6
• Remark 3.1
• Example 3.2
• Example 3.3
• Example 3.4