Shor \([[9,1,3]]\) code[1]
Description
Nine-qubit CSS code that is the smallest such code to correct a single-qubit error. The logical state is encoded using \begin{align} \begin{split} |\overline{0}\rangle&=\frac{1}{2\sqrt{2}}\left(|000\rangle+|111\rangle\right)^{\otimes3}\\ |\overline{1}\rangle&=\frac{1}{2\sqrt{2}}\left(|000\rangle-|111\rangle\right)^{\otimes3}~. \end{split} \end{align} The code works by concatenating each qubit of a phase-flip with a bit-flip repetition code. Therefore, the code can correct both type of errors simultaneously.
Specifically, a state is phase-flip error-corrected by a three-qubit phase-flip repetition code, with stabilizer generators \(X_0 X_1I_2\) and \(X_0I_1X_2\) in \(X\) basis, where the subscript represents the qubit index. Each logical qubit is encoded using \begin{align} \label{eq:phase-flip} \begin{split} |\overline{0}\rangle &= |+_0+_1+_2\rangle \\ |\overline{1}\rangle &= |-_0-_1-_2\rangle . \end{split} \end{align} Then, each physical qubit used in \eqref{eq:phase-flip} is further encoded in the three-qubit bit-flip repetition code, \begin{align} |\pm _j \rangle = \frac{1}{\sqrt{2}}( |0_{j0}0_{j1}0_{j2}\rangle \pm |1_{j0}1_{j1}1_{j2}\rangle), \end{align} each with bit-flip error stabilizer generators \(Z_{j0}Z_{j1}I_{j2}\) and \(Z_{j0}I_{j1}Z_{j2} \) with \(j=0,1,2\). Notice now the phase-flip error stabilizer generator is extended as \(X_j = X_{j0}X_{j1}X_{j2}\). As a result, the stabilizer generators with the qubit index flattened are \begin{align} \begin{split} Z_{j0}Z_{j1}I_{j2} &\rightarrow \{Z_0Z_1, Z_3Z_4, Z_6Z_7\} \\ Z_{j0}I_{j1}Z_{j2} &\rightarrow \{Z_0Z_2, Z_3Z_5, Z_6Z_8\} \\ X_0 X_1I_2 &\rightarrow \{X_0X_1X_2X_3X_4X_5\}\\ X_0 I_1X_2 &\rightarrow \{X_0X_1X_2X_6X_7X_8\}. \end{split} \end{align}
Protection
Realizations
Notes
Parent
- Quantum parity code (QPC) — Shor's code is part of the sub-family of \([[m^2,1,m]]\) QPC codes.
Cousins
- Quantum repetition code — Shor's code is a concatenation of a three-qubit bit-flip with a three-qubit phase-flip repetition code.
- Concatenated quantum code — Shor's code is a concatenation of a three-qubit bit-flip with a three-qubit phase-flip repetition code.
- Projective-plane surface code — Shor's code is the smallest surface code defined on the projective plane [5].
Zoo code information
References
- [1]
- P. W. Shor, “Scheme for reducing decoherence in quantum computer memory”, Physical Review A 52, R2493 (1995). DOI
- [2]
- N. H. Nguyen et al., “Demonstration of Shor Encoding on a Trapped-Ion Quantum Computer”, Physical Review Applied 16, (2021). DOI; 2104.01205
- [3]
- D. M. Debroy et al., “Optimizing Stabilizer Parities for Improved Logical Qubit Memories”, Physical Review Letters 127, (2021). DOI; 2105.05068
- [4]
- R. Zhang et al., “Loss-tolerant all-photonic quantum repeater with generalized Shor code”, Optica 9, 152 (2022). DOI; 2203.07979
- [5]
- Michael H. Freedman and David A. Meyer, “Projective plane and planar quantum codes”. quant-ph/9810055
Cite as:
“Shor \([[9,1,3]]\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/shor_nine
Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/small/Shor_nine.yml.