\([[9,1,3]]\) 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 \\ I & I & I & X & X & X & X & X & X \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 using reinforcement learning [3].Fault-tolerant measurement-free logical-zero state preparation [4].Decoding
Bit- and phase-flip circuits utilize CNOT and Hadamard gates ([5], Fig. 10.6).Fault Tolerance
Fault-tolerant logical zero and logical plus state preparation using reinforcement learning [3].Fault-tolerant measurement-free logical-zero state preparation [4].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 [6]. Variants of the code to handle coherent noise studied and realized by K. Brown and C. Monroe groups [7].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% [8]. All-photonic quantum repeater architecture tested on the same code [9].Cousins
- Quantum repetition code— The Shor code is a concatenation of a three-qubit bit-flip with a three-qubit phase-flip repetition code.
- Concatenated qubit code— The Shor code is a concatenation of a three-qubit bit-flip with a three-qubit phase-flip repetition code.
- \([[9,1,4,3]]\) Nine-qubit Bacon-Shor code— The \([[9,1,4,3]]\) Bacon-Shor code reduces to the Shor code for a particular gauge configuration.
- Quantum error-correcting code (QECC)— The Shor code is the first quantum error-correcting code.
- Cluster-state code— The Shor code admits a codeword that is the cluster state of a particular nine-vertex graph [10,11].
- \([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code— The Lloyd-Slotine nine-mode code is a bosonic analogue of Shor's code.
- Hybrid stabilizer code— The Shor code can be modified to store three additional classical bits to yield a \([[9,1:3,3]]\) hybrid stabilizer code [12].
- \(((9,2,3))\) Ruskai code— The \(((9,2,3))\) Ruskai code results from projecting the Shor code into the PI qubit subspace [13].
- \([[8,2,2]]\) hyperbolic color code— The Shor code (\([[8,2,2]]\) color code) is a small surface (color) code defined on the projective plane.
- \([[9,1,3]]\) Surface-17 code— Both Shor's code and surface-17 are \([[9,1,3]]\) codes, but they are distinct (e.g., they have different quantum weight enumerators).
Member of code lists
- 2D stabilizer codes
- Concatenated quantum codes and friends
- Hamiltonian-based codes
- Quantum codes
- Quantum codes based on homological products
- Quantum codes with fault-tolerant gadgets
- Quantum codes with notable decoders
- Quantum CSS codes
- Quantum LDPC codes
- Realized quantum codes
- Small-distance quantum codes and friends
- Stabilizer codes
- Surface code and friends
- Topological codes
Primary Hierarchy
Parents
The Shor code is part of the sub-family of \([[m^2,1,m]]\) QPCs.
Projective-plane surface codeKitaev surface CDSC Twist-defect surface Lattice stabilizer Homological Generalized homological-product QLDPC CSS Stabilizer Qubit Abelian topological Topological Hamiltonian-based QECC Quantum
The Shor code is one of the nine-qubit surface codes defined on the projective plane \(\mathbb{R}P^2\) [14; Fig. 4][15; Fig. 20].
\([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit codeLattice stabilizer Generalized homological-product QLDPC CSS Stabilizer Abelian topological Topological Hamiltonian-based Concatenated quantum Small-distance block quantum QECC Quantum
The \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code for \(q=2\) reduces to the \([[9,1,3]]\) Shor code.
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[9,1,3]]\) Shor code
References
- [1]
- P. W. Shor, “Scheme for reducing decoherence in quantum computer memory”, Physical Review A 52, R2493 (1995) DOI
- [2]
- A. B. Khesin, J. Z. Lu, and P. W. Shor, “Universal graph representation of stabilizer codes”, (2025) 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]
- H. Goto, Y. Ho, and T. Kanao, “Measurement-free fault-tolerant logical-zero-state encoding of the distance-three nine-qubit surface code in a one-dimensional qubit array”, Physical Review Research 5, (2023) arXiv:2303.17211 DOI
- [5]
- M. Nakahara, “Quantum Computing”, (2008) DOI
- [6]
- 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
- [7]
- 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
- [8]
- 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
- [9]
- 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
- [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
- [14]
- M. H. Freedman and D. A. Meyer, “Projective plane and planar quantum codes”, (1998) arXiv:quant-ph/9810055
- [15]
- H. Bombin and M. A. Martin-Delgado, “Homological error correction: Classical and quantum codes”, Journal of Mathematical Physics 48, (2007) arXiv:quant-ph/0605094 DOI
Page edit log
- Remmy Zen (2024-07-15) — most recent
- Victor V. Albert (2022-06-29)
- Victor V. Albert (2022-03-15)
- Victor V. Albert (2021-12-10)
- Qingfeng (Kee) Wang (2021-12-07)
Cite as:
“\([[9,1,3]]\) Shor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/shor_nine