Bivariate bicycle code[1] 


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 are \(n\) \(X\) and \(Z\) check operators, with each one of weight six.


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 \(k/2n\). For example, the \([[144,12,12]]\) code has ancilla-added rate \(1/24\). In contrast, the distance-13 surface code has ancilla-added rate \(1/338\).


Syndrome extraction circuit requires seven layers of CNOT gates regardless of code length. BP-OSD decoder [2] has been extended [1] to account for measurement errors (i.e., the circuit-based noise model [3]).Random and optimized syndrome extraction schedules from Ref. [1] are not distance preserving.

Fault Tolerance

Fault-tolerant state initialization using Tanner graph techniques [4] and an ancillary surface code [1].


\(0.8\%\) pseudothreshold for circuit-level noise under BP-OSD decoder [1] (cf. [3]).



  • Kitaev surface code — 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.


S. Bravyi et al., “High-threshold and low-overhead fault-tolerant quantum memory”, (2023) arXiv:2308.07915
P. Panteleev and G. Kalachev, “Degenerate Quantum LDPC Codes With Good Finite Length Performance”, Quantum 5, 585 (2021) arXiv:1904.02703 DOI
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
L. Z. Cohen et al., “Low-overhead fault-tolerant quantum computing using long-range connectivity”, Science Advances 8, (2022) arXiv:2110.10794 DOI
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Zoo Code ID: qcga

Cite as:
“Bivariate bicycle code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.
@incollection{eczoo_qcga, title={Bivariate bicycle code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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Cite as:

“Bivariate bicycle code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.