\([[9,1,4,3]]\) Nine-qubit Bacon-Shor code

\([[9,1,4,3]]\) Nine-qubit Bacon-Shor code[1,2]

Description

Error-correcting nine-qubit subsystem stabilizer code encoding one logical qubit and four gauge qubits. There are exactly four inequivalent CSS gauge fixings of the code, including the Shor code and the surface-17 code [3].

The stabilizer subgroup is generated by \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*{(1)}\end{align} while a convenient generating set for the gauge group is completed by the eight weight-two gauge operators \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*{(2)}\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.

Decoding

Message passing for \([[9,1,4,3]]\) Bacon-Shor code [4].

Code Capacity Threshold

\(2.02 \times 10^{-5}\) concatenated threshold for the recursively concatenated code [5].

Realizations

Trapped-ion qubits: state preparation, logical measurement, and syndrome extraction (deferred to the end) demonstrated on a 13-qubit device by M. Cetina and C. Monroe groups [6].Neutral atom arrays: repeated error correction demonstrated on a device by Atom Computing [7].

Cousins

\([[9,1,3]]\) Shor code— The Shor code is a CSS gauge fixing of the nine-qubit Bacon-Shor code [3,8].
\([[9,1,3]]\) Surface-17 code— The \([[9,1,3]]\) rotated surface code is a CSS gauge fixing of the nine-qubit Bacon-Shor code [3].
Small-distance qubit stabilizer code

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Subsystem qubit codeSubsystem QECC Quantum

Subsystem qubit stabilizer codeSubsystem QECC Quantum

Subsystem CSS codeSubsystem QECC Quantum

Bravyi-Bacon-Shor (BBS) code

Bacon-Shor codeSubsystem qubit Lattice subsystem Subsystem QECC Quantum

Parents

Bacon-Shor code

The nine-qubit Bacon-Shor code is the shortest error-correcting Bacon-Shor code.

\([[9,1,4,3]]\) Nine-qubit Bacon-Shor code

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) arXiv:quant-ph/0506023 DOI
[3]: A. Cross and D. Vandeth, “Small Binary Stabilizer Subsystem Codes”, (2025) arXiv:2501.17447
[4]: Z. W. E. Evans and A. M. Stephens, “Message passing in fault-tolerant quantum error correction”, Physical Review A 78, (2008) arXiv:0806.2188 DOI
[5]: F. M. Spedalieri and V. P. Roychowdhury, “Latency in local, two-dimensional, fault-tolerant quantum computing”, (2008) arXiv:0805.4213
[6]: L. Egan et al., “Fault-Tolerant Operation of a Quantum Error-Correction Code”, (2021) arXiv:2009.11482
[7]: B. W. Reichardt et al., “Fault-tolerant quantum computation with a neutral atom processor”, (2025) arXiv:2411.11822
[8]: D. Poulin, “Stabilizer Formalism for Operator Quantum Error Correction”, Physical Review Letters 95, (2005) arXiv:quant-ph/0508131 DOI

Page edit log

Victor V. Albert (2024-12-12) — most recent

Cite as:

“\([[9,1,4,3]]\) Nine-qubit Bacon-Shor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/bacon_shor_9

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/subsystem/qldpc/bbs/bacon_shor/bacon_shor_9.yml.