\([[9,1,3]]\) Shor code[1] 

Description

Nine-qubit CSS code that is the first quantum error-correcting code.

Logical codewords are \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} \tag*{(1)}\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} \tag*{(2)}\end{align} Then, each physical qubit used in (2) 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), \tag*{(3)}\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} \tag*{(4)}\end{align}

Protection

The code detects two-qubit errors or corrects an arbitrary single-qubit error.

Decoding

Bit- and phase-flip circuits utilize CNOT and Hadamard gates ([2], Fig. 10.6).

Realizations

Trapped-ion qubits: state preparation with 98.8(1)% and 98.5(1)% fidelity for state \(|\overline{0}\rangle\) and \(|\overline{1}\rangle\), respectively, by N. Linke group [3]. Variants of the code to handle coherent noise studied and realized by K. Brown and C. Monroe groups [4].Optical systems: quantum teleportation of information implemented by J.-W. Pan group on maximally entangled pair of one physical and one logical qubit with fidelity rate of up to 78.6% [5]. All-photonic quantum repeater architecture tested on the same code [6].

Parents

Cousins

References

[1]
P. W. Shor, “Scheme for reducing decoherence in quantum computer memory”, Physical Review A 52, R2493 (1995) DOI
[2]
M. Nakahara, “Quantum Computing”, (2008) DOI
[3]
N. H. Nguyen et al., “Demonstration of Shor Encoding on a Trapped-Ion Quantum Computer”, Physical Review Applied 16, (2021) arXiv:2104.01205 DOI
[4]
D. M. Debroy et al., “Optimizing Stabilizer Parities for Improved Logical Qubit Memories”, Physical Review Letters 127, (2021) arXiv:2105.05068 DOI
[5]
Y.-H. Luo et al., “Quantum teleportation of physical qubits into logical code spaces”, Proceedings of the National Academy of Sciences 118, (2021) arXiv:2009.06242 DOI
[6]
R. Zhang et al., “Loss-tolerant all-photonic quantum repeater with generalized Shor code”, Optica 9, 152 (2022) arXiv:2203.07979 DOI
[7]
M. H. Freedman and D. A. Meyer, “Projective plane and planar quantum codes”, (1998) arXiv:quant-ph/9810055
[8]
Griffiths, Robert B. "Graph states and graph codes."
[9]
Y. Hwang and J. Heo, “On the relation between a graph code and a graph state”, (2015) arXiv:1511.05647
[10]
M. B. Ruskai, “Pauli Exchange Errors in Quantum Computation”, Physical Review Letters 85, 194 (2000) arXiv:quant-ph/9906114 DOI
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Zoo Code ID: shor_nine

Cite as:
\([[9,1,3]]\) Shor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/shor_nine
BibTeX:
@incollection{eczoo_shor_nine, title={\([[9,1,3]]\) Shor code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/shor_nine} }
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Cite as:

\([[9,1,3]]\) Shor 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/edit/main/codes/quantum/qubits/small_distance/small/shor_nine.yml.