Generalised Bicycle Codes QLDPC

• Generalised bicycle codes are CSS QLDPC codes defined via pairs of circulant matrices or group-algebra elements, ensuring automatic commutativity and robust construction.
• They interpolate between toric, hypergraph-product, and classical bicycle codes, achieving distance scaling of Θ(√n) for bounded-weight stabilizers.
• Their hardware-friendly design offers reduced qubit overhead and efficient decoding strategies, making them promising for scalable quantum computing.

Generalised bicycle codes (GB codes) are a canonical family of CSS quantum low-density parity-check (QLDPC) codes constructed via pairs of commuting binary circulant matrices or, more generally, as two-block group-algebra codes over finite (typically cyclic or abelian) groups. These codes interpolate between classical bicycle codes, toric (surface) codes, hypergraph-product codes, and more general group-theoretic CSS constructions. They are of current research interest due to their favorable rate, qubit efficiency, and distance properties, as well as their amenability to efficient decoding and hardware implementation.

1. Algebraic Construction and Parameters

A generalised bicycle code is a CSS code defined by two binary $\ell\times\ell$ circulant matrices $A$ and $B$, or more generally group-algebra elements in $\mathbb F_2[G]$ for finite group $G$ (typically $G = \mathbb Z_\ell$ for basic GB codes). The stabilizer check matrices are

$H_X = [A \mid B], \qquad H_Z = [B^T \mid A^T]$

with $n = 2\ell$ qubits and each check having weight equal to the total number of nonzero coefficients in the corresponding circulant polynomials $a(x), b(x)$. CSS commutativity is automatic: $AB = BA$.

The quantum code parameters are:

• Length: $n=2\ell$
• Dimension: $$k = 2\,\deg \operatorname{gcd}(a(x), b(x), x^\ell-1)$$
• Minimum distance: $d \geq \min(d_A, d_B)$, where $d_A$ and $d_B$ are the distances of the classical codes with parity checks $A, B$ respectively. For general $w$ (row weight), the best attainable minimum distance for CSS-GB QLDPC codes is $d = \Theta(\sqrt{n})$ for typical families (Wang et al., 2022), and codes with $d = O(n)$ exist only at vanishing rate or for sporadic lengths.

In the group-algebra formulation, one has $H_X = [L(a)|R(b)]$ with $L(a)$ (left-regular) and $R(b)$ (right-regular) matrices over $\mathbb F_2[G]$; dimension and distance formulas extend with the same structure for arbitrary (possibly non-abelian) $G$ (Lin et al., 2023).

2. Distance Bounds and Code Families

Lower Bounds

For bounded check weight $w$, the best possible scaling for minimum distance is $d = O(n^{1/2})$ due to D-dimensional locality bounds (Bravyi-Poulin-Terhal): a CSS code local in $D$ dimensions satisfies $d = O(n^{1-1/D})$ and, for $w=4$, $D=3$ so $d = O(n^{1/2})$ (Wang et al., 2022).

Exact families achieving optimal distance include:

• $(2,2)$-GB codes: For $A(x)=1+x$, $B(x)=1+x^\alpha$, the distance satisfies

$d \geq \lambda(L)$

where $L$ is a sublattice of $\mathbb Z^2$ related to the offsets of $A$ and $B$ (Arnault et al., 28 Jul 2025). For specific choices, $[[2n^2,2,n]]$, $[[4r^2,2,2r]]$, $[[(2t+1)^2+1,2,2t+1]]$ families exactly saturate the surface code distance bounds, with even-distance cases breaking a long-held belief that only odd distances were possible for GB codes (Arnault et al., 28 Jul 2025).

• Families with higher check weight ($w=6,8$) exhibit $d \sim A(w)\sqrt{n}+B(w)$ with $A$ increasing in $w$, but the scaling remains $\Theta(\sqrt{n})$ (Wang et al., 2022).

Small-length and High-rate Codes

For small lengths ($n<200$) exhaustive enumerations yield many new codes with distance exceeding the toric code at the same length for weight-4 stabilizers (Arnault et al., 25 Apr 2025, Arnault et al., 28 Jul 2025). Examples include codes with $2n=5, d=4$, $2n=13, d=8$, $2n=33, d=12$—all record-setting for their parameters. For higher weights, codes such as $[[54,6,9]]$, $[[96,8,12]]$ have been constructed (Lin et al., 2023).

3. Generalisations: Group-Algebra and Bivariate Codes

GB codes are special cases of two-block group algebra (2BGA) codes, where the circulant group $C_n$ is replaced by an arbitrary finite group $G$ (possibly non-abelian) (Lin et al., 2023, Pacenti et al., 15 Sep 2024). In the group-algebra framework, $A = \sum_{s\in S_1}s , B = \sum_{t\in S_2}t$, with $S_1, S_2$ subsets of $G$.

Bivariate bicycle (BB) codes extend GB codes by using polynomials in two commuting variables $x,y$ over $R=\mathbb F_2[x, y]/(x^\ell-1, y^m-1)$, yielding codes on $n = 2\ell m$ qubits, and offer further structural flexibility and improved rates in some settings (Symons et al., 17 Nov 2025, Shaw et al., 23 Jul 2024).

ZSZ codes generalize BB codes to non-abelian groups (e.g., $G=\mathbb Z_\ell\rtimes_q\mathbb Z_m$), achieving improved passive-decoding thresholds (Guo et al., 29 Jul 2025).

4. Syndrome Extraction, Decoding, and Robustness

GB codes have naturally redundant stabilizer generators—each circulant yields $\ell$ (not all independent) rows—enabling enhanced measurement error protection. The syndrome code forms a classical code with distance $d_S$ (often $d_S=3$ for weight-3 generators), protecting against multiple measurement errors per round (Lin et al., 26 Feb 2025).

Efficient decoding is achieved using belief propagation (BP) on the factor graph associated to the CSS code, optionally augmented by ordered-statistics decoding (OSD) or MBP$_4$/ADOSD$_4$. Single-shot or two-shot decoding leverages syndrome redundancy for fast and robust error correction, achieving logical error rates near full multi-round performance but at reduced latency (Lin et al., 26 Feb 2025, Mostad et al., 9 May 2025).

Finite-length simulations show that suitably designed GB codes (potentially with higher or unconstrained row weight) outperform or match quantum Tanner and product codes under practical noise models, with competitive logical error rates and no observed error floors down to $10^{-7}$ for $n=144$–512 (Mostad et al., 9 May 2025). Row weights up to 8–13 are compatible with current superconducting- or atom-based hardware.

Fault-tolerant syndrome extraction circuits for GB and BB codes can be implemented with CNOT-depth as low as 6–7 (weight-3 codes), and with qubit connectivities reduced to as low as degree 5 via morphing-circuit constructions, while retaining error performance (Shaw et al., 23 Jul 2024).

5. Resource Overhead and Implementation

GB codes yield a per-logical-qubit overhead of $O(d)$, where $d$ is the code distance, contrasting the $O(d^2)$ overhead of surface codes. For instance, a $[[126,12,10]]$ code achieves $N_\text{total}/k = 38$ versus $400$ for surface code at $d=10$—a reduction by an order of magnitude, and this advantage grows with $d$ (Webster et al., 20 Nov 2025).

On neutral atom arrays, efficient movement and measurement protocols allow 2–3$\times$ faster logical cycles and 5–10$\times$ smaller physical footprints than equivalent surface-code memories (Viszlai et al., 2023). The atom array implementation entails organizing data and check qubits in interleaved grids and co-moving check ancillas for syndrome extraction via global Rydberg pulses.

In quantum memory hierarchies combining GB and surface codes, overall spacetime volume for computational benchmarks can be reduced by factors as large as $5\times$ compared to surface-code-only approaches (Viszlai et al., 2023).

6. Extensions and Theoretical Frameworks

GB codes embed as the $c=1$ case of hyperbicycle codes, which interpolate between the bicycle family and hypergraph-product codes. In these models, the CSS code is generated via superpositions of "block" permutation matrices, yielding a continuum of tradeoffs in rate and distance scaling (Kovalev et al., 2012). For group-algebraic codes, non-abelian instantiations such as ZSZ and quantum Margulis codes further widen the accessible parameter space, including codes with improved passive-decoding capabilities and logarithmic distance scaling in special cases (Guo et al., 29 Jul 2025, Pacenti et al., 15 Sep 2024).

Graph covering constructions extend BB codes to infinite sequences with predictable parameter scaling. When a cover code is an $h$-fold cover of a base code, one obtains $n_h = hn$, $k_h \geq k$, and $d \leq d_h \leq h d$ (conjectured to always hold) (Symons et al., 17 Nov 2025).

7. Applications, Performance, and Outlook

Generalised bicycle codes play a central role in the search for scalable, low-overhead QLDPC codes for practical quantum computing. Their structural flexibility permits hardware-adapted scheduling, efficient logical gate gadgets, and implementation in settings requiring high encoding rates or reduced qubit degree. The current literature demonstrates order-of-magnitude qubit savings, competitive thresholds ($\approx 0.5\%$ under circuit noise), and practical decoding protocols suitable for both small and moderate code lengths (Webster et al., 20 Nov 2025, Mostad et al., 9 May 2025, Lin et al., 26 Feb 2025, Viszlai et al., 2023).

Open research directions include rigorous threshold proofs for all code parameter regimes, systematic passive and single-shot decoding for non-abelian generalizations, and further extension of graph covering and algebraic embedding techniques to maximize code performance and experimental feasibility across hardware platforms.