Bivariate bicycle (BB) code

Bivariate bicycle (BB) code[1]

Description

One of several Abelian 2BGA codes which admit time-optimal syndrome measurement circuits that can be implemented in a two-layer architecture, a generalization of the square-lattice architecture optimal for the surface codes. Codes can be classified by the weight of their checks, e.g., by BB\(w\) where \(w\) is the check weight.

The qubit connectivity graph is not quite a 2D grid and is instead decomposable into two planar subgraphs of degree three; there exists an optimized layout minimizing Euclidean communication distance for check operators [2]. There are \(n\) \(X\) and \(Z\) check operators, with each one of weight six.

See Refs. [3,4] for examples of self-dual BB codes. Several variants and generalizations exist [5,6]. There exist qudit BB codes that achieve \(kd^2 / n = 20\) [7].

Protection

Admits an \(0.8\%\) pseudo-threshold for circuit-level noise under BP-OSD decoder [1] (cf. [8]).

Rate

When ancilla qubit overhead is included, the encoding rate surpasses that of the surface code. A general \([[n,k,d]]\) bivariate bicycle code requires \(n\) ancilla qubits for encoding, meaning that its ancilla-added encoding rate is \(k/2n\).

Transversal Gates

Logical Pauli operators and fold-transversal gates studied in Ref. [9,10].

Decoding

Syndrome extraction circuit requires seven layers of CNOT gates regardless of code length. BP-OSD decoder [11] has been extended [1] to account for measurement errors (i.e., the circuit-based noise model [8]).The depth-7 syndrome extraction schedules studied in Ref. [1] are not generally distance-preserving; for the \([[144,12,12]]\) code, all \(936\) depth-7 variants obtained by reordering the CNOT layers satisfy \(d_{\mathrm{circ}}\leq 10 < d\).Some long-range check operators can be measured less frequently than others [12].Syndrome extraction circuits called morphing circuits [13], generalizing circuits for the color code [14].Decoding under circuit-level noise has been studied for the BP, BP+OSD, and AutDEC decoders [15].Transformer-based neural-network decoder [16].Matching decoder [17].

Fault Tolerance

Fault-tolerant state initialization using lattice surgery techniques [18,19] and an ancillary surface code [1].

Realizations

Superconducting circuits: syndrome extraction has been implemented for the \([[18,4,4]]\) BB code on a 32-qubit Kunlun device by the Wang, Song, and Deng groups [20]. The same is also shown for an \([[18,6,3]]\) code obtained by removing two check operators from the former code [20].

Notes

A database of bivariate bicycle codes is available at QECDB by S. Burton.

Cousins

Honeycomb (6.6.6) color code— Certain bivariate bicycle codes are equivalent to a family of 6.6.6 color codes [21].
Abelian topological code— BB codes have been investigated in terms of their anyons and topological order [22].
Generalized bicycle (GB) code— GB codes (BB codes) are 2BGA codes over the cyclic group \(\mathbb{Z}_{\ell}\) (Abelian group \(\mathbb{Z}_{r} \times \mathbb{Z}_{s}\)). The two codes are the same when \(r\) and \(s\) are relatively prime due to the isomorphism \(\mathbb{Z}_{r} \times \mathbb{Z}_{s} \cong \mathbb{Z}_{\ell = rs}\).

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum

Parents

Generalized homological-product qubit CSS code

Two-block group-algebra (2BGA) codesGeneralized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum

Bivariate bicycle codes are Abelian 2BGA codes over groups of the form \(\mathbb{Z}_{r} \times \mathbb{Z}_{s}\).

Abelian LP codeGeneralized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum

Bivariate bicycle codes are Abelian LP codes over groups of the form \(\mathbb{Z}_{r} \times \mathbb{Z}_{s}\).

2D lattice stabilizer codeLattice stabilizer Stabilizer Hamiltonian-based QECC Quantum

Bivariate bicycle codes are defined on 2D lattices with periodic boundary conditions, and versions with open boundary conditions have been investigated [21,23]. Bivariate bicycle codes are on par with the surface code in terms of threshold, but admit a much higher ancilla-added encoding rate at the expense of having non-geometrically local weight-six check operators. BB codes have been investigated in terms of their anyons and topological order [22].

Bivariate bicycle (BB) code

Children

\([[108,8,10]]\) BB6 code

\([[288,12,18]]\) double-gross code

BB5 code

\([[72,12,6]]\) BB6 code

\([[90,8,10]]\) BB6 code

\([[144,12,12]]\) gross code

References

[1]: S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, “High-threshold and low-overhead fault-tolerant quantum memory”, Nature 627, 778 (2024) arXiv:2308.07915 DOI
[2]: C. Poole, T. M. Graham, M. A. Perlin, M. Otten, and M. Saffman, “Architecture for fast implementation of quantum low-density parity-check codes with optimized Rydberg gates”, Physical Review A 111, (2025) arXiv:2404.18809 DOI
[3]: Z. Liang and Y.-A. Chen, “Self-dual bivariate bicycle codes with transversal Clifford gates”, (2026) arXiv:2510.05211
[4]: Q. Xu, H. Zhou, D. Bluvstein, M. Cain, M. Kalinowski, J. Preskill, M. D. Lukin, and N. Maskara, “Batched high-rate logical operations for quantum LDPC codes”, (2025) arXiv:2510.06159
[5]: L. Voss, S. J. Xian, T. Haug, and K. Bharti, “Multivariate Bicycle Codes”, (2025) arXiv:2406.19151
[6]: M. Wang and F. Mueller, “Coprime Bivariate Bicycle Codes and Their Layouts on Cold Atoms”, Quantum 10, 2009 (2026) arXiv:2408.10001 DOI
[7]: Z. Liang and Y.-A. Chen, “Generalized \(\mathbb{Z}_p\) toric codes as qudit low-density parity-check codes”, (2026) arXiv:2602.20158
[8]: A. G. Fowler, A. M. Stephens, and P. Groszkowski, “High-threshold universal quantum computation on the surface code”, Physical Review A 80, (2009) arXiv:0803.0272 DOI
[9]: J. N. Eberhardt and V. Steffan, “Logical Operators and Fold-Transversal Gates of Bivariate Bicycle Codes”, (2024) arXiv:2407.03973
[10]: H. Sayginel, S. Koutsioumpas, M. Webster, A. Rajput, and Dan E Browne, “Fault-Tolerant Logical Clifford Gates from Code Automorphisms”, (2025) arXiv:2409.18175
[11]: P. Panteleev and G. Kalachev, “Degenerate Quantum LDPC Codes With Good Finite Length Performance”, Quantum 5, 585 (2021) arXiv:1904.02703 DOI
[12]: N. Berthusen, D. Devulapalli, E. Schoute, A. M. Childs, M. J. Gullans, A. V. Gorshkov, and D. Gottesman, “Toward a 2D Local Implementation of Quantum Low-Density Parity-Check Codes”, PRX Quantum 6, (2025) arXiv:2404.17676 DOI
[13]: M. H. Shaw and B. M. Terhal, “Lowering Connectivity Requirements for Bivariate Bicycle Codes Using Morphing Circuits”, Physical Review Letters 134, (2025) arXiv:2407.16336 DOI
[14]: C. Gidney and C. Jones, “New circuits and an open source decoder for the color code”, (2023) arXiv:2312.08813
[15]: S. Koutsioumpas, H. Sayginel, M. Webster, and Dan E Browne, “Automorphism Ensemble Decoding of Quantum LDPC Codes”, (2025) arXiv:2503.01738
[16]: J. Blue, H. Avlani, Z. He, L. Ziyin, and I. L. Chuang, “Machine Learning Decoding of Circuit-Level Noise for Bivariate Bicycle Codes”, (2025) arXiv:2504.13043
[17]: K. Sahay, D. J. Williamson, and B. J. Brown, “A matching decoder for bivariate bicycle codes”, (2026) arXiv:2602.22770
[18]: L. Z. Cohen, I. H. Kim, S. D. Bartlett, and B. J. Brown, “Low-overhead fault-tolerant quantum computing using long-range connectivity”, Science Advances 8, (2022) arXiv:2110.10794 DOI
[19]: Q. Xu, J. P. B. Ataides, C. A. Pattison, N. Raveendran, D. Bluvstein, J. Wurtz, B. Vasic, M. D. Lukin, L. Jiang, and H. Zhou, “Constant-Overhead Fault-Tolerant Quantum Computation with Reconfigurable Atom Arrays”, (2023) arXiv:2308.08648
[20]: K. Wang et al., “Demonstration of low-overhead quantum error correction codes”, Nature Physics (2026) arXiv:2505.09684 DOI
[21]: J. N. Eberhardt, F. R. F. Pereira, and V. Steffan, “Pruning qLDPC codes: Towards bivariate bicycle codes with open boundary conditions”, (2024) arXiv:2412.04181
[22]: Anonymous, “Anyon theory and topological frustration of high-efficiency quantum low-density parity-check codes”, Physical Review Letters (2025) arXiv:2503.04699 DOI
[23]: Z. Liang, B. Yang, J. T. Iosue, and Y.-A. Chen, “Operator algebra and algorithmic construction of boundaries and defects in (2+1)D topological Pauli stabilizer codes”, (2025) arXiv:2410.11942

Page edit log

Mark Webster (2025-03-25) — most recent
Victor V. Albert (2025-03-25)
Victor V. Albert (2023-10-16)
Leonid Pryadko (2023-10-10)
Victor V. Albert (2023-10-04)

Cite as:

“Bivariate bicycle (BB) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/qcga

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/qcga.yml.