\([[9,1,3]]\) 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} A stabilizer tableau for the code is \begin{align} \begin{bmatrix} Z & Z & I & I & I & I & I & I & I \\ I & Z & Z & I & I & I & I & I & I \\ I & I & I & Z & Z & I & I & I & I \\ I & I & I & I & Z & Z & I & I & I \\ I & I & I & I & I & I & Z & Z & I \\ I & I & I & I & I & I & I & Z & Z \\ X & X & X & X & X & X & I & I & I \\ X & X & X & X & X & X & I & I & I \end{bmatrix}~. \tag*{(2)}\end{align} The encoder-respecting form of the Shor code is a star-shaped tree graph [2]. 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.

Protection

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

Encoding

Fault-tolerant logical zero and logical plus state preparation [3].

Decoding

Bit- and phase-flip circuits utilize CNOT and Hadamard gates ([4], Fig. 10.6).

Fault Tolerance

Fault-tolerant logical zero and logical plus state preparation [3].

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 [5]. Variants of the code to handle coherent noise studied and realized by K. Brown and C. Monroe groups [6].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% [7]. All-photonic quantum repeater architecture tested on the same code [8].

Parents

Cousins

References

[1]
P. W. Shor, “Scheme for reducing decoherence in quantum computer memory”, Physical Review A 52, R2493 (1995) DOI
[2]
J. Z. Lu, A. B. Khesin, and P. W. Shor, “Universal graph representation of stabilizer codes”, (2024) arXiv:2411.14448
[3]
R. Zen, J. Olle, L. Colmenarez, M. Puviani, M. Müller, and F. Marquardt, “Quantum Circuit Discovery for Fault-Tolerant Logical State Preparation with Reinforcement Learning”, (2024) arXiv:2402.17761
[4]
M. Nakahara, “Quantum Computing”, (2008) DOI
[5]
N. H. Nguyen, M. Li, A. M. Green, C. Huerta Alderete, Y. Zhu, D. Zhu, K. R. Brown, and N. M. Linke, “Demonstration of Shor Encoding on a Trapped-Ion Quantum Computer”, Physical Review Applied 16, (2021) arXiv:2104.01205 DOI
[6]
D. M. Debroy, L. Egan, C. Noel, A. Risinger, D. Zhu, D. Biswas, M. Cetina, C. Monroe, and K. R. Brown, “Optimizing Stabilizer Parities for Improved Logical Qubit Memories”, Physical Review Letters 127, (2021) arXiv:2105.05068 DOI
[7]
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
[8]
R. Zhang, L.-Z. Liu, Z.-D. Li, Y.-Y. Fei, X.-F. Yin, L. Li, N.-L. Liu, Y. Mao, Y.-A. Chen, and J.-W. Pan, “Loss-tolerant all-photonic quantum repeater with generalized Shor code”, Optica 9, 152 (2022) arXiv:2203.07979 DOI
[9]
M. H. Freedman and D. A. Meyer, “Projective plane and planar quantum codes”, (1998) arXiv:quant-ph/9810055
[10]
Griffiths, Robert B. "Graph states and graph codes."
[11]
Y. Hwang and J. Heo, “On the relation between a graph code and a graph state”, (2015) arXiv:1511.05647
[12]
I. Kremsky, M.-H. Hsieh, and T. A. Brun, “Classical enhancement of quantum-error-correcting codes”, Physical Review A 78, (2008) arXiv:0802.2414 DOI
[13]
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.), 2024. https://errorcorrectionzoo.org/c/shor_nine
BibTeX:
@incollection{eczoo_shor_nine, title={\([[9,1,3]]\) Shor code}, booktitle={The Error Correction Zoo}, year={2024}, 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.), 2024. https://errorcorrectionzoo.org/c/shor_nine

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/shor_nine.yml.