# Bravyi-Bacon-Shor (BBS) code[1]

Also known as Generalized Bacon-Shor code.

## Description

An \([[n,k,d]]\) CSS subsystem stabilizer code generalizing Bacon-Shor codes to a larger set of qubit geometries. Defined through a binary matrix \(A\) such that physical qubits live on sites \((i,j)\) whenever \(A_{i,j}=1\). The gauge group is generated by 2-qubit operators, including \(XX\) interations between any two qubits sharing a column in \(A\), and \(ZZ\) interations between two qubits sharing a row. The code parameters are: \(n=\sum_{i,j}A_{i,j}\), \(k=\text{rank}(A)\), and the distance is the minimum weight of any row or column.

## Protection

Detects errors on \(d-1\) qubits, corrects errors on \(\left\lfloor (d-1)/2 \right\rfloor\) qubits, where \(d\) is the minimum weight of a row or column in \(A\) [2].

## Rate

## Parent

## Child

## Cousins

- Subsystem hypergraph product (SHP) code — The BBS code construction can utilize different classical codes in different rows and columns of \(A\), while the subsystem construction does not; see [1; pg. 4]
- Subsystem hypergraph product (SHP) code — SHP and BBS codes have been numerically compared [2].

## References

- [1]
- S. Bravyi, “Subsystem codes with spatially local generators”, Physical Review A 83, (2011) arXiv:1008.1029 DOI
- [2]
- M. Li and T. J. Yoder, “A Numerical Study of Bravyi-Bacon-Shor and Subsystem Hypergraph Product Codes”, (2020) arXiv:2002.06257
- [3]
- T. J. Yoder, “Optimal quantum subsystem codes in two dimensions”, Physical Review A 99, (2019) arXiv:1901.06319 DOI

## Page edit log

- Victor V. Albert (2022-01-20) — most recent
- Srilekha Gandhari (2021-12-13)

## Cite as:

“Bravyi-Bacon-Shor (BBS) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/bravyi_bacon_shor