Breaking the bicycle frame: Coset-based quantum LDPC codes

Abstract: Generalizing the construction of two-block group algebra (2BGA) codes, we introduce a family of two-block quantum LDPC codes constructed using the action of a group on the cosets of its subgroup. This replaces the regular group actions of the earlier two-block constructions and significantly expands the search space, yielding new quantum LDPC codes outside the 2BGA family. Through a computer search, we identify several new quantum LDPC codes, including weight-6 codes with parameters $[[48,8,6]]$, $[[96,8,10]]$, and $[[224,12,16]]$, as well as weight-8 codes with parameters $[[84,16,8]]$, $[[112,16,10]]$, $[[128,16,12]]$, and $[[168,16,15]]$. Furthermore, we introduce a maximally packed syndrome extraction schedule of depth $w+2$, including initialization and measurement steps, for any code with a maximum stabilizer weight of $w$ from our family. Under a standard circuit-level noise model, our codes, when decoded using BP-OSD, perform competitively with BB codes, achieving thresholds of $\approx0.65\%$ for the weight-6 family and $\approx0.35\%$ for the weight-8 family. Finally, we introduce a group-theoretic framework to generate sequences of graph-based covers of 2BGA codes, recovering and extending recent results on code constructions of this type.

• The paper introduces a coset-based framework that generalizes two-block group algebra codes, yielding new quantum LDPC codes with improved parameters.
• It leverages arbitrary left and right group actions on coset spaces to design Tanner graphs with controlled thickness and optimized syndrome extraction circuits.
• Computational searches reveal codes with lower stabilizer weights and significantly reduced logical error rates, enhancing hardware efficiency for fault-tolerant quantum computing.

Coset-Based Quantum LDPC Codes: A Generalization Beyond Two-Block Group Algebra Constructions

Introduction

The development of quantum low-density parity-check (LDPC) codes with practical parameters is crucial for scalable quantum information processing and fault-tolerant architectures. While advances have produced asymptotically good quantum LDPC code families, such as product and Tanner codes, much of the focus for hardware implementation has remained on structures with planar or near-planar connectivity, exemplified by surface codes and bivariate bicycle (BB) codes. The present work introduces a broadened group-theoretic construction paradigm: coset-based quantum LDPC codes, which strictly generalize two-block group algebra (2BGA) codes and thus subsume BB, generalized bicycle (GB), trivariate bicycle, and related product-based constructions. This extension leverages group actions on coset spaces to provide a substantially larger code design space, leading to new quantum LDPC codes unattainable within the 2BGA framework.

Theoretical Framework

The central construction pivots from using left and right regular group actions (as in 2BGA codes) to arbitrary left and right actions of a finite group $G$ on the cosets of a subgroup $H \leq G$. Given the normalizer $N_G(H)$, the left actions $L: G \rightarrow S_{G/H}$ and right actions $R:N_G(H)\rightarrow S_{G/H}$ commute, ensuring the CSS orthogonality required for quantum codes. The code family is parameterized by selecting group-algebra elements $a\in\mathbb{F}[G]$ and $b\in\mathbb{F}[N_G(H)]$, forming parity check matrices:

$$H_X = [L(a) \mid R(b)], \quad H_Z = [-R(b)^T \mid L(a)^T]$$

This formalism recovers all 2BGA codes when $H$ is normal in $G$. However, by exploring arbitrary (including non-normal) subgroups, the construction yields codes with parameter sets and Tanner graph structures outside the reach of existing group-algebraic approaches.

Structural Implications for Tanner Graphs and Syndrome Extraction

A significant contribution of this work is a combinatorial and graph-theoretic analysis of the Tanner graphs generated by these coset-based codes. A key parameter is the graph "thickness", corresponding to the minimal number of planar layers required for circuit realization. For codes with regular stabilizer weights, tight upper and lower bounds on thickness are provided, and it is shown that for codes with stabilizer weight $H \leq G$0 and $H \leq G$1, the thicknesses are exactly 2 and 3, respectively, matching those of BB codes and enabling efficient mapping onto hardware with layered local connectivity constraints.

Furthermore, maximally packed syndrome extraction circuits are described. The protocol achieves minimal time-domain depth—$H \leq G$2 time steps per syndrome round including initialization and measurement—for any code of stabilizer weight $H \leq G$3, generalizing and strictly improving upon prior art for BB and cyclic HGP codes.

Computational Search and Explicit New Codes

A systematic computational search over group/subgroup pairs—focusing on non-abelian $H \leq G$4 and non-normal $H \leq G$5—identified several new quantum LDPC codes with nontrivial parameters, including:

• Weight-6 codes: $H \leq G$6, $H \leq G$7, $H \leq G$8
• Weight-8 codes: $H \leq G$9, $N_G(H)$0, $N_G(H)$1, $N_G(H)$2

These codes are not equivalent to previously reported BB/2BGA codes, as verified by direct enumeration and comparison (for lengths up to 100). Notably, codes such as $N_G(H)$3 are of strictly lower stabilizer weight than the best 2BGA codes of identical length. The search methodology effectively utilizes group and subgroup conjugacy arguments to minimize redundancy.

Fault-Tolerant Performance and Decoding

Extensive Monte Carlo simulations were performed under the standard circuit-level noise model, using the BP-OSD decoder as well as the relay-BP and beam search decoders. Key results include:

• Weight-6 family: pseudo-threshold $N_G(H)$4; e.g., $N_G(H)$5 achieves logical error rates $N_G(H)$6 at $N_G(H)$7.
• Weight-8 family: pseudo-threshold $N_G(H)$8, but higher encoding rates and larger code distances; $N_G(H)$9 achieves $L: G \rightarrow S_{G/H}$0 at $L: G \rightarrow S_{G/H}$1.

The relay-BP decoder achieves substantial reductions in logical error rate—up to 327-fold compared to BP-OSD for some codes—demonstrating that these codes are compatible with fast, practical, and hardware-efficient decoding strategies.

Graph Cover Lifts and Families

The group-theoretic framework admits the systematic construction of sequences of larger codes as lifts (covers) of a fixed base code via normal subgroup extensions. This yields families where code parameters (rate, distance, stabilizer weight) can be tuned systematically by selecting base group/subgroup pairs and their covering group extensions. The methodology strictly generalizes prior cover constructions for BB codes and accommodates non-abelian structures. Computational search over cover codes identifies large blocklength codes with previously unattained parameters, e.g., a triple cover $L: G \rightarrow S_{G/H}$2 of a GB code $L: G \rightarrow S_{G/H}$3.

Implications and Outlook

The coset-based framework dramatically expands the design landscape for quantum LDPC codes, especially for finite, hardware-relevant blocklengths. It subsumes all known group-algebraic constructions, provides new codes with superior parameters and lower stabilizer weight, and enables efficient, scalable syndrome extraction protocols. The group-theoretic perspective on cover graphs facilitates the design of code families suitable for message-passing decoders and enables analytical exploration of logical operator structure and distance scaling.

Future Research Directions

• Hardware mapping: Tailoring code and coset selection to specific physical layouts (e.g., 2D chip or neutral atom graphs) for improved syndrome extraction fidelity and efficiently engineering planar or low-layer implementations.
• Logical structure analysis: Characterizing logical operators and minimum-weight undetectable errors for coset-based codes with non-normal $L: G \rightarrow S_{G/H}$4.
• Distance scaling and thresholds in lifted families: Providing rigorous bounds on minimum distance for covers, including non-abelian lift families, leveraging the generalized group-theoretic formalism.

Conclusion

Coset-based quantum LDPC codes generalize and enhance the two-block group algebra framework, yielding new quantum codes with enhanced parameters, tractable syndrome extraction, and robust performance under hardware-relevant noise models and scalable decoders (2606.17268). Group actions on coset spaces constitute a powerful algebraic toolkit, facilitating further advances in quantum error correction theory and implementation.