## Description

Subsystem CSS code defined on an \(m_1 \times m_2\) lattice of qubits that generalizes the \([[9,1,3]]\) (subspace) Shor code. It is said to be symmetric when \(m_1=m_2\), and asymmetric otherwise.

The \(X\)-type and \(Z\)-type stabilizers defined as \(X\) and \(Z\) operators acting on all qubits on adjacent columns and rows, respectively. Let \(O_{i,j}\) denote an operator acting on the qubit at a position \((i,j)\) on the lattice, with \(i\in\{0,1,\ldots ,m_1-1\}\) and \(j\in\{0,1,\ldots,m_2-1\}\). The code's stabilizer group is \begin{align} \mathsf{S}=\langle X_{i,*}X_{i+1,*},Z_{*,j}Z_{*,j+1}\rangle~, \tag*{(1)}\end{align} with generators expressed as products of nearest-neightbour 2-qubit gauge operators, \begin{align} \begin{split} X_{i,*}X_{i+1,*}= \bigotimes_{k=0}^{m_2-1} X_{i,k}X_{i+1,k} \\ Z_{*,j}Z_{*,j+1}=\bigotimes_{k=0}^{m_1-1} Z_{k,j}Z_{k,j+1}~. \end{split} \tag*{(2)}\end{align} Syndrome extraction can be done by measuring these gauge operators, which are on fewer qubits and local.

The shortest error-correcting Bacon-Shor code is \([[9,1,3,3]]\), with four stabilizer generators \begin{align} \begin{array}{ccccccccc} X & X & X & X & X & X & I & I & I\\ I & I & I & X & X & X & X & X & X\\ Z & Z & I & Z & Z & I & Z & Z & I\\ I & Z & Z & I & Z & Z & I & Z & Z \end{array}~, \tag*{(3)}\end{align} which generate the gauge group with the help of eight additional generators \begin{align} \begin{array}{ccccccccc} X & I & I & X & I & I & I & I & I\\ I & X & I & I & X & I & I & I & I\\ I & I & I & X & I & I & X & I & I\\ I & I & I & I & X & I & I & X & I\\ Z & Z & I & I & I & I & I & I & I\\ I & I & I & Z & Z & I & I & I & I\\ I & Z & Z & I & I & I & I & I & I\\ I & I & I & I & Z & Z & I & I & I \end{array}~. \tag*{(4)}\end{align} If the physical qubits are arranged in a three-by-three square, the \(Z\)-type (\(X\)-type) gauge operators are supported on qubits in the same row (column). The code reduces to the Shor code for a particular gauge configuration.

## Protection

## Rate

## Transversal Gates

## Gates

## Decoding

## Fault Tolerance

## Code Capacity Threshold

## Threshold

## Realizations

## Notes

## Parents

- Bravyi-Bacon-Shor (BBS) code
- Subsystem hypergraph product (SHP) code
- Compass code — A compass code on a fully non-colored lattice reduces to the Bacon-Shor code.

## Cousins

- Hamiltonian-based code — The 2D Bacon-Shor gauge-group Hamiltonian is the compass model [23–25]. Bacon-Show code Hamiltonians can be used to suppress errors in adiabatic quantum computation [26], while subspace-code two-local Hamiltonians cannot [27].
- Hastings-Haah Floquet code — The Bacon-Shor code admits a Floquet version with a particular stabilizer measurement schedule [28].
- Asymmetric quantum code — Bacon-Shor code parameters against bit- and phase-noise can be optimized by changing the block geometry, yielding good performance against biased noise [3]. A fault-tolerant teleportation-based computation scheme for asymmetric Bacon-Shor codes is effective against highly biased noise [15].
- Self-correcting quantum code — 3D Bacon-Shor codes were conjectured to be self-correcting [2], but there remain issues to be resolved in order to validate this conjecture (see [29; Sec. IX.B]).
- GNU PI code — GNU codes of length \((2t+1)^2\) result from projecting Bacon-Shor codes into the PI qubit subspace [30].
- \([[9,1,3]]\) Shor code — The \([[9,1,3,3]]\) Bacon-Shor code reduces to the Shor code for a particular gauge configuration.
- \([[4,2,2]]\) CSS code — The error-detecting \([[4,1,1,2]]\) Bacon-Shor code, which reduces to a subcode of the \([[4,2,2]]\) code for a particular gauge configuration, has gauge operators \(\{XXII,IIXX,ZIZI,IZIZ\}\).
- Quantum parity code (QPC) — Bacon-Shor codes reduce to QPCs when all \(X\)-type gauge generators are fixed [31; pg. 6].
- Heavy-hexagon code — Bacon-Shor stabilizers are used to measure the X-type stabilizers of the code.

## References

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- [2]
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- [31]
- M. Li et al., “2D Compass Codes”, Physical Review X 9, (2019) arXiv:1809.01193 DOI

## Page edit log

- Victor V. Albert (2022-07-31) — most recent
- Mazin Karjikar (2022-06-28)
- Victor V. Albert (2022-03-15)
- Srilekha Gandhari (2022-01-20)
- Victor V. Albert (2021-12-03)

## Cite as:

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