Bivariate Bicycle Codes in Quantum LDPC

Updated 17 October 2025

Bivariate Bicycle Codes are defined by two-variable polynomial constructions over finite rings that generalize classical cyclic codes to robust quantum LDPC architectures.
They provide high encoding rates with constant-weight stabilizers and support transversal Clifford gates for fault-tolerant implementations.
Advanced decoder methods, connectivity engineering, and hardware experiments highlight their potential for scalable, resource-efficient quantum computers.

Bivariate Bicycle Codes are a family of quantum low-density parity-check (qLDPC) codes defined by algebraic and group-theoretic constructions over two variables—reflecting an underlying two-dimensional lattice symmetry with periodic or twisted boundary conditions. These codes generalize classical cyclic and bicycle codes to the bivariate case, supporting high encoding rates, constant-weight stabilizers, and the potential for fault-tolerant implementations with transversal Clifford gates. They form the foundation of recent advances in scalable, low-overhead quantum error correction architectures and have been the subject of analytical, numerical, and experimental research efforts.

1. Algebraic Construction and Group Structure

Bivariate bicycle codes are formulated within the CSS (Calderbank–Shor–Steane) stabilizer framework, defined using two polynomials $A(x, y)$ and $B(x, y)$ over the ring $\mathbb{F}_2[x, y]/(x^\ell - 1, y^m - 1)$ . The physical qubits are associated with pairs in the group $\mathbb{Z}_\ell \times \mathbb{Z}_m$ , and the parity-check matrices are given by

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

guaranteeing the CSS condition via $H_X H_Z^\top = AB + BA = 0$ . Code families can be systematically constructed by choosing sparse polynomial generators such as $A(x, y) = x^{a_1} + y^{a_2} + y^{a_3}$ and $B(x, y) = y^{b_1} + x^{b_2} + x^{b_3}$ , with code existence conditions typically requiring odd, coprime $\ell$ and $m$ so that the ring is semisimple and the code structure can be mapped via $z = xy$ to a univariate ideal in $\mathbb{F}_2[z]/(z^{\ell m}-1)$ (Postema et al., 24 Feb 2025, Wang et al., 19 Aug 2024).

Logical operators and code parameters are fully characterized in this algebraic language. For coprime BB codes (with $\gcd(\ell, m) = 1$ ), the dimension is determined by the greatest common divisor (gcd) of the univariate reductions, $k = 2 \cdot \deg(h(z))$ where $h(z) = \gcd(a(z), b(z), z^{\ell m} - 1)$ . Minimum distance $d$ is computed as the minimal weight of representatives of nontrivial logical classes. Ring-theoretic and homological algebra methods, including Koszul complexes, also clarify the relationship between anyonic excitations and logical operator spaces (Chen et al., 6 Mar 2025, Eberhardt et al., 4 Jul 2024).

2. Parameter Scaling, Code Rate, and Asymptotic Badness

Bivariate bicycle codes are notable for their high encoding rates, often allowing $k / n$ (the rate) to exceed that of surface and color codes for moderate code sizes. Empirically, codes such as $[[72, 12, 6]]$ , $[[144, 12, 12]]$ , and $[[360, 12, 24]]$ have been identified, realizing $k d^2 / n$ ratios significantly exceeding those of conventional topological codes—e.g., $k d^2 / n \sim 9.6$ for $[[120,8,12]]$ and even higher for other codes on twisted tori (Liang et al., 5 Mar 2025, Liang et al., 6 Oct 2025).

However, in the asymptotic limit, BB codes with fixed stabilizer weight are "asymptotically bad": as $n \rightarrow \infty$ , the relative distance $d/n$ and rate $k/n$ vanish ( $d/n \leq O(n^{-1/5})$ , $k/n \leq O(d^{-1/2})$ ). Thus, while these codes offer compelling performance and overhead reduction for moderate $n \lesssim 1000$ , they are not candidates for quantum LDPC code constructions with asymptotically nonzero rate and distance (Postema et al., 24 Feb 2025).

3. Stabilizer Locality, Twisted Tori, and Self-Dual Codes

BB codes' stabilizer generators are constant-weight (e.g., weight 6 for standard BB codes, weight 8 for self-dual constructions). The geometric locality of stabilizers—critical for physical implementation—is enhanced by defining codes on "twisted" tori, corresponding to nontrivial lattice vectors such as $(a_1 = (0, m), a_2 = (l, q))$ . These choices lead to better code distances and improved stabilizer locality: for instance, the $[[360,12,24]]$ code can be realized with minimal-extent (weight-6) stabilizers using a $(3,3)$ -BB code on a twisted torus, thereby simplifying syndrome extraction circuits (Liang et al., 5 Mar 2025).

Self-dual BB codes, where stabilizers respect an antipodal symmetry (e.g., $f(x, y)$ and its inversion $\overline{f(x, y)}$ ), support transversal implementations of the full Clifford group (CNOT, Hadamard, and $S$ gates), provided stabilizer weight is doubly even and duality is respected. This makes self-dual BB codes direct generalizations and sometimes strict improvements over color codes, offering higher rates and efficient transversal logic (Liang et al., 6 Oct 2025).

4. Decoding Strategies and Post-Selection

Several families of decoders have been developed for BB codes, addressing the challenges posed by their highly symmetric, densely cycled Tanner graphs. Standard belief-propagation (BP) decoders can be improved via ensemble methods such as AutDEC, which leverages the large automorphism group of BB codes to permute syndromes and aggregate results from parallel BP decoders, achieving logical error rates comparable to BP+OSD-0 but with lower time overhead (Koutsioumpas et al., 3 Mar 2025).

Tesseract, a search-based ML decoder, finds most-likely errors via A*-pruned search and exploits specific error structure to improve efficiency and robustness, yielding up to $14\times$ — $19\times$ physical qubit savings over surface codes for the $[[144,12,12]]$ BB code at moderate error rates (Beni et al., 14 Mar 2025). Recent work demonstrates that recurrent transformer-based machine learning decoders can outperform BP-OSD for small BB codes, reducing logical error rates while maintaining constant inference time (Blue et al., 17 Apr 2025).

Cluster-based post-selection strategies applied to BB codes further improve effective logical error rates. Heuristic confidence metrics based on error cluster size and log-likelihood ratio statistics allow for post-selection with moderate abort rates (e.g., three orders of magnitude logical error suppression at $\approx$ 1–19% abort rates for the $[[144,12,12]]$ code at $p=0.1$ – $0.3\%$ ) (Lee et al., 7 Oct 2025).

5. Hardware Implementation, Overhead, and Connectivity Engineering

BB codes have been implemented experimentally—for example, a distance-4 $[[18,4,4]]$ BB code demonstrated on a 32-qubit superconducting processor with long-range couplers, achieving logical error rates per cycle competitive with or better than surface code patches for equivalent distances and qubit budgets (Wang et al., 14 May 2025). The codes' natural embedding in a regular or biplanar structure enables efficient syndrome extraction: the stabilizer measurement circuit can run in low depth (e.g., 7 CZ layers alternating with single-qubit gates), and the encoding rate (up to $1/8$ in the above experiment) significantly reduces qubit overhead relative to planar codes.

Connectivity requirements—initially degree 6—can be relaxed using morphing circuits or advanced routing frameworks such as Louvre. Morphing circuits reduce connectivity to degree 5 by alternating measurement cycles and contracting stabilizer supports, preserving performance and facilitating biplanar layouts for logical I/O with ancillary codes (Shaw et al., 23 Jul 2024). Louvre-based routing reuses couplers and inserts SWAP/CXSWAP layers to further decrease the required degree (to 4.5 or 4 in Louvre-7/8), with only a minor penalty to the logical error rate (Zhou et al., 28 Aug 2025). These techniques enable implementation on planar or modular chip architectures with constrained wiring.

6. Generalizations, Topological Order, and Fault-Tolerance

The topological order and anyonic structure underlying BB codes have been analytically characterized. The codes instantiate $\mathbb{Z}_2^Q$ gauge theory, with ground-state degeneracy computed by the quotient ring dimension; explicit formulas for logical dimension are derived using Gröbner bases and the Bernstein–Khovanskii–Kushnirenko theorem for the mixed volume of Newton polytopes (Chen et al., 6 Mar 2025). Phenomena such as topological frustration and quasi-fractonic anyon mobility, as well as symmetry-enriched topological (SET) orders, arise naturally and influence the scaling of logical qubits with system size.

Fold-transversal logic gates—Hadamard, phase, and SWAP gates acting on pairs related by geometric or algebraic duality—can be constructed in symmetric (self-dual) BB code variants, extending the set of fault-tolerant operations available beyond surface and color codes (Eberhardt et al., 4 Jul 2024, Liang et al., 6 Oct 2025).

Extensions to non-abelian group structures, such as ZSZ codes based on the $\mathbb{Z}_\ell \rtimes \mathbb{Z}_m$ semidirect product, enable further improvements in autonomous error correction thresholds and support scalable, self-correcting memory architectures for platforms with movable tweezer arrays (Guo et al., 29 Jul 2025).

7. Real-World Architectures and Resource-Efficient Computation

The modular "bicycle architecture" directly integrates BB codes as the quantum memory and computational backbone, supporting explicit logical instruction sets—including idle cycles, shift automorphisms, in-module and inter-module Pauli measurements, and T-injection protocols. Compilation strategies map arbitrary quantum algorithms into measurement-based sequences compatible with syndrome extraction cycles and LPU (logical processing unit) gadgets. Resource estimates demonstrate %%%%45 $\mathbb{F}_2[z]/(z^{\ell m}-1)$ 46%%%% improvements in maximum logical circuit depth versus surface code architectures for equal numbers of physical qubits in several regimes. These improvements stem from both the high rate and the optimized instruction set enabled by the code structure (Yoder et al., 3 Jun 2025).

Photonic fusion-based implementations also benefit from the LDPC nature, high rate, and nonlocality that BB codes accommodate. Foliated BB codes implemented by fusing deterministic resource states (branched chains and star-graphs/GHZ states) produced by quantum emitters can achieve error-correction thresholds on par with toric codes, with the added advantage of higher encoding rates. Robustness is further ensured by advanced RUS (repeat-until-success) protocols and tailored union-find decoding for photon erasure and fusion failures (Chen et al., 21 Sep 2025).

In summary, bivariate bicycle codes represent a class of quantum LDPC codes with rich algebraic and topological structure, offering high encoding rates, constant-weight local stabilizers, and compatibility with transversal Clifford gates. While asymptotically not LDPC-optimal, for moderate scale ( $n \lesssim 1000$ ), they deliver qubit efficiency and logical error rates highly competitive with or superior to conventional surface codes and color codes. Their architecture supports scalable, modular, resource-efficient quantum computers implemented across superconducting, neutral atom, and photonic platforms, benefiting from both constructional flexibility and advanced decoder technologies.