## Description

CSS subsystem stabilizer code defined on an \(m_1 \times m_2\) lattice of qubits. It is said to be symmetric when \(m_1=m_2\). 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~, \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} \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 \([[9,1,3]]\) and has 6 gauge operators, symmetric in both \(X\) and \(Z\), reducing to the Shor code for a particular gauge configuration. The error-detecting \([[4,1,2]]\) Bacon-Shor code, which reduces to a subcode of the \([[4,2,2]]\) code for a particular gauge configuration, has gauge operators \(\{XIXI,IIXX,ZIZI,IZIZ\}\).

## Protection

## Rate

## Transversal Gates

## Gates

## Decoding

## Fault Tolerance

## Threshold

## Realizations

## Parent

## Cousins

- GNU permutation-invariant code — Symmetrized versions of the Bacon-Shor codes are GNU codes
- Heavy-hexagon code — Bacon-Shor stabilizers are used to measure the X-type stabilizers of the code.
- Quantum parity code (QPC) — Bacon-Shor codes reduce to QPCs for a particular gauge configuration.

## Zoo code information

## References

- [1]
- P. W. Shor, “Scheme for reducing decoherence in quantum computer memory”, Physical Review A 52, R2493 (1995). DOI
- [2]
- D. Bacon, “Operator quantum error-correcting subsystems for self-correcting quantum memories”, Physical Review A 73, (2006). DOI; quant-ph/0506023
- [3]
- P. Aliferis and A. W. Cross, “Subsystem Fault Tolerance with the Bacon-Shor Code”, Physical Review Letters 98, (2007). DOI; quant-ph/0610063
- [4]
- X. Zhou, D. W. Leung, and I. L. Chuang, “Methodology for quantum logic gate construction”, Physical Review A 62, (2000). DOI; quant-ph/0002039
- [5]
- Theodore J. Yoder, “Universal fault-tolerant quantum computation with Bacon-Shor codes”. 1705.01686
- [6]
- Yoder, Theodore., DSpace@MIT Practical Fault-Tolerant Quantum Computation (2018)
- [7]
- Matthew B. Hastings, Jeongwan Haah, and Ryan O'Donnell, “Fiber Bundle Codes: Breaking the $N^{1/2} \operatorname{polylog}(N)$ Barrier for Quantum LDPC Codes”. 2009.03921
- [8]
- M. B. Hastings and J. Haah, “Dynamically Generated Logical Qubits”, Quantum 5, 564 (2021). DOI; 2107.02194
- [9]
- Laird Egan et al., “Fault-Tolerant Operation of a Quantum Error-Correction Code”. 2009.11482

## Cite as:

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