Lifts of quantum CSS codes

TL;DR

This work introduces a general notion of lifting quantum CSS codes through the Tanner cone-complex, yielding a unified geometric framework for generating longer codes while preserving low-weight checks. By exploiting finite coverings, the authors connect lifts to the fiber-bundle and balanced-product perspectives and provide a complete classification of HPC lifts via Goursat quintuples, showing equivalence with lifted product codes. They also present three new families of liftable, non-product codes (E- and V-type) derived from presentation complexes, and demonstrate through numerical experiments that lifts can achieve improved relative parameters (kd^2/n) at moderate lengths. The framework offers a versatile toolkit for constructing LDPC quantum codes with potentially good asymptotics and practical fault-tolerant implementations, while pointing to future work on non-regular lifts and deeper parameter analyses.

Abstract

We propose a notion of lift for quantum CSS codes, inspired by the geometrical construction of Freedman and Hastings. It is based on the existence of a canonical complex associated to any CSS code, that we introduce under the name of Tanner cone-complex, and over which we generate covering spaces. As a first application, we describe the classification of lifts of hypergraph product codes (HPC) and demonstrate the equivalence with the lifted product code (LPC) of Panteleev and Kalachev, including when the linear codes, factors of the HPC, are Tanner codes. As a second application, we report several new non-product constructions of quantum CSS codes, and we apply the prescription to generate their lifts which, for certain selected covering maps, are codes with improved relative parameters compared to the initial one.

Figures (9)

• Figure 1: Tanner graph of Steane's 7 qubit code Steane1996. The upper row of vertices represents $Z$-checks, while the middle and lower rows are respectively qubits and $X$-checks. The induced subgraph of a $Z$-check $z$ is colored in blue.
• Figure 2: Cone on an induced subgraph $\mathcal{T}_z\hookrightarrow T(C_X)$, obtained by attaching $\mathcal{T}_z\times I$ to $\mathcal{T}(C_X)$ on one end, and then collapsing the other end to a point labeled $z$. Thick lines are edges of $\mathcal{T}_z$, dots are qubit vertices and boxes are check vertices. Dashed lines represent other edges in $\mathcal{T}(C_X)$. Thin lines are edges in $\mathcal{T}_z\times I$ or $\operatorname{C}\mathcal{T}_z$.
• Figure 3: Presentation complex of $M_{2,3}$.
• Figure 4: Square cellulation $M_{2,3}^\Box$ corresponding to $\operatorname{EL}(2,3,1)$. Qubits are on the edges, while $X$-checks on the vertices and $Z$-checks on the faces. Non-labeled cells can be inferred from the others. Red edges represent the support of a minimal weight logical $X$-operator, while blue edges represent the support of a minimal weight logical $Z$-operator.
• Figure 5: Square cellulation $M^\Box_{2,3}$ corresponding to $\operatorname{VL}(2,3,1)$. Qubits are on the vertices, alternating with $X$-checks and $Z$-checks. Each vertex labeled $q,x,z$ represents a different basis vector. Non labeled vertices can be inferred from labeled ones.

Theorems & Definitions (83)

• Definition 2.1: CSS code
• Proposition 2.2
• Definition 2.3: Hypergraph product code
• Lemma 2.4
• Theorem 2.5: Hurewicz
• Proposition 2.6
• Theorem 2.7: Galois correspondence
• Lemma 2.8
• Definition 2.9: Lift of a graph
• Remark 2.10