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

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

Description

Error-correcting nine-qubit subsystem stabilizer code encoding one logical qubit and three gauge qubits.

Admits the following stabilizers, \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} which generate the gauge group with the help of eight additional gauge-group 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*{(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 \([[9,1,3,3]]\) Bacon-Shor code [3].

Code Capacity Threshold

\(2.02 \times 10^{-5}\) concatenated threshold for the recursively concatenated code [4].

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 [5].Rydberg atomic devices: repeated error correction demonstrated on a device by Atom Computing [6].

Parent

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

Cousins

Small-distance block quantum code
\([[9,1,3]]\) Shor code — The \([[9,1,3,3]]\) Bacon-Shor code reduces to the Shor code for a particular gauge configuration.

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]: 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
[4]: F. M. Spedalieri and V. P. Roychowdhury, “Latency in local, two-dimensional, fault-tolerant quantum computing”, (2008) arXiv:0805.4213
[5]: L. Egan et al., “Fault-Tolerant Operation of a Quantum Error-Correction Code”, (2021) arXiv:2009.11482
[6]: B. W. Reichardt et al., “Logical computation demonstrated with a neutral atom quantum processor”, (2024) arXiv:2411.11822

Page edit log

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

Cite as:

“\([[9,1,3,3]]\) 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.