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.
Protection
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].
Decoding
Syndrome extraction circuit requires seven layers of CNOT gates regardless of code length. BP-OSD decoder [7] 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 [8].Syndrome extraction circuits called morphing circuits [9], generalizing circuits for the color code [10].
Fault Tolerance
Fault-tolerant state initialization using lattice surgery techniques [11,12] and an ancillary surface code [1].
Parents
- Generalized homological-product qubit CSS code
- Two-block group-algebra (2BGA) codes — Bivariate bicycle codes are Abelian 2BGA codes over groups of the form \(\mathbb{Z}_{r} \times \mathbb{Z}_{s}\).
- Abelian LP code — Bivariate bicycle codes are Abelian LP codes over groups of the form \(\mathbb{Z}_{r} \times \mathbb{Z}_{s}\).
- Quantum LDPC (QLDPC) code
Child
Cousin
- 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.
References
- [1]
- S. Bravyi et al., “High-threshold and low-overhead fault-tolerant quantum memory”, Nature 627, 778 (2024) arXiv:2308.07915 DOI
- [2]
- C. Poole et al., “Architecture for fast implementation of qLDPC codes with optimized Rydberg gates”, (2024) arXiv:2404.18809
- [3]
- L. Voss et al., “Multivariate Bicycle Codes”, (2024) arXiv:2406.19151
- [4]
- M. Wang and F. Mueller, “Coprime Bivariate Bicycle Codes and their Properties”, (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]
- P. Panteleev and G. Kalachev, “Degenerate Quantum LDPC Codes With Good Finite Length Performance”, Quantum 5, 585 (2021) arXiv:1904.02703 DOI
- [8]
- N. Berthusen et al., “Toward a 2D Local Implementation of Quantum LDPC Codes”, (2024) arXiv:2404.17676
- [9]
- M. H. Shaw and B. M. Terhal, “Lowering Connectivity Requirements For Bivariate Bicycle Codes Using Morphing Circuits”, (2024) arXiv:2407.16336
- [10]
- C. Gidney and C. Jones, “New circuits and an open source decoder for the color code”, (2023) arXiv:2312.08813
- [11]
- L. Z. Cohen et al., “Low-overhead fault-tolerant quantum computing using long-range connectivity”, Science Advances 8, (2022) arXiv:2110.10794 DOI
- [12]
- Q. Xu et al., “Constant-Overhead Fault-Tolerant Quantum Computation with Reconfigurable Atom Arrays”, (2023) arXiv:2308.08648
Page edit log
- Victor V. Albert (2023-10-16) — most recent
- 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.), 2023. https://errorcorrectionzoo.org/c/qcga