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.

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.

Several variants and generalizations exist [3,4].

Protection

Admits an \(0.8\%\) pseudo-threshold for circuit-level noise under BP-OSD decoder [1] (cf. [5]).

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. [6,7].

Decoding

Syndrome extraction circuit requires seven layers of CNOT gates regardless of code length. BP-OSD decoder [8] has been extended [1] to account for measurement errors (i.e., the circuit-based noise model [5]).Random and optimized syndrome extraction schedules from Ref. [1] are not distance-preserving.Some long-range check operators can be measured less frequently than others [9].Syndrome extraction circuits called morphing circuits [10], generalizing circuits for the color code [11].Decoding under circuit-level noise has been studied for the BP, BP+OSD, and AutDEC decoders [12].Transformer-based neural-network decoder [13].

Fault Tolerance

Fault-tolerant state initialization using lattice surgery techniques [14,15] 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 [16]. The same is also shown for an \([[18,6,3]]\) code obtained by removing two check operators from the former code [16].

Cousins

Honeycomb (6.6.6) color code— Certain bivariate bicycle codes are equivalent to a family of 6.6.6 color codes [17].
Balanced product (BP) code— The \([[90,8,10]]\) bivariate bicycle code can be formulated as a balanced product of two cyclic codes [18].
Abelian topological code— BB codes have been investigated in terms of their anyons and topological order [19].

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 Qubit Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

Parents

Generalized homological-product qubit CSS code

Two-block group-algebra (2BGA) codesGeneralized homological-product QLDPC 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 codeCSS Generalized homological-product QLDPC 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 QLDPC 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 [17,20]. 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 [19].

Bivariate bicycle (BB) code

Children

BB5 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 qLDPC codes with optimized Rydberg gates”, (2024) arXiv:2404.18809
[3]: L. Voss, S. J. Xian, T. Haug, and K. Bharti, “Multivariate Bicycle Codes”, (2025) arXiv:2406.19151
[4]: M. Wang and F. Mueller, “Coprime Bivariate Bicycle Codes”, (2024) arXiv:2408.10001
[5]: 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
[6]: J. N. Eberhardt and V. Steffan, “Logical Operators and Fold-Transversal Gates of Bivariate Bicycle Codes”, (2024) arXiv:2407.03973
[7]: H. Sayginel, S. Koutsioumpas, M. Webster, A. Rajput, and Dan E Browne, “Fault-Tolerant Logical Clifford Gates from Code Automorphisms”, (2025) arXiv:2409.18175
[8]: P. Panteleev and G. Kalachev, “Degenerate Quantum LDPC Codes With Good Finite Length Performance”, Quantum 5, 585 (2021) arXiv:1904.02703 DOI
[9]: 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
[10]: 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
[11]: C. Gidney and C. Jones, “New circuits and an open source decoder for the color code”, (2023) arXiv:2312.08813
[12]: S. Koutsioumpas, H. Sayginel, M. Webster, and Dan E Browne, “Automorphism Ensemble Decoding of Quantum LDPC Codes”, (2025) arXiv:2503.01738
[13]: 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
[14]: 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
[15]: 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
[16]: K. Wang et al., “Demonstration of low-overhead quantum error correction codes”, (2025) arXiv:2505.09684
[17]: 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
[18]: R. Tiew and N. P. Breuckmann, “Low-Overhead Entangling Gates from Generalised Dehn Twists”, (2024) arXiv:2411.03302
[19]: K. Chen, Y. Liu, Y. Zhang, Z. Liang, Y.-A. Chen, K. Liu, and H. Song, “Anyon Theory and Topological Frustration of High-Efficiency Quantum LDPC Codes”, (2025) arXiv:2503.04699
[20]: 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”, (2024) 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/homological/balanced_product/lp/qcga.yml.