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

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

## Decoding

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].

## Threshold

## Parents

- Qubit CSS code
- Two-block group-algebra (2BGA) codes — Bivariate bicycle codes are Abelian 2BGA codes over groups of the form \(\mathbb{Z}_{r} \times \mathbb{Z}_{s}\).
- Quantum low-density parity-check (QLDPC) code

## 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”, (2023) arXiv:2308.07915
- [2]
- P. Panteleev and G. Kalachev, “Degenerate Quantum LDPC Codes With Good Finite Length Performance”, Quantum 5, 585 (2021) arXiv:1904.02703 DOI
- [3]
- 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
- [4]
- L. Z. Cohen et al., “Low-overhead fault-tolerant quantum computing using long-range connectivity”, Science Advances 8, (2022) arXiv:2110.10794 DOI

## 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 code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/qcga