\([[9,1,3]]\) Surface-17 code[1]
Alternative names: \([[9,1,3]]\) rotated surface code.
Description
A \([[9,1,3]]\) rotated surface code named for the sum of its 9 data qubits and 8 syndrome qubits. It is one of the four inequivalent CSS gauge fixings of the nine-qubit Bacon-Shor code [2]. It uses the smallest number of qubits to perform fault-tolerant error correction on a surface code with parallel syndrome extraction.
A stabilizer tableau for the code is given by [3; ID 8519]. \begin{align} \begin{array}{ccccccccc} I & X & I & I & I & I & I & X & I \\ I & I & I & X & X & I & I & I & I \\ I & I & X & I & I & X & X & X & I \\ X & I & I & X & I & X & I & I & X \\ Z & I & I & I & I & I & I & I & Z \\ I & I & Z & I & I & I & Z & I & I \\ I & I & I & Z & Z & Z & Z & I & I \\ I & Z & I & I & I & Z & I & Z & Z \end{array}~. \tag*{(1)}\end{align} The code is depicted in Fig. I.
Protection
Independent correction of single-qubit \(X\) and \(Z\) errors. Correction for some two-qubit \(X\) and \(Z\) errors. Admits pseudo-thresholds of \(\approx 10^{-4}\) under depolarizing noise.Encoding
Measurement-free fault-tolerant logical zero state preparation in nearest-neighbor qubit connectivity [4].Fault-tolerant logical zero and logical plus state preparation in all-to-all connectivity, and fault-tolerant logical zero state preparation on 2D grids, with flag qubits [5].Transversal Gates
Pauli gates, CNOT gate, and \(H\) gate (with relabeling).Fault Tolerance
Measurement-free fault-tolerant logical zero state preparation in nearest-neighbor qubit connectivity [4].Fault-tolerant logical zero and logical plus state preparation in all-to-all connectivity, and fault-tolerant logical zero state preparation on 2D grids, with flag qubits [5].Realizations
Implemented at ETH Zurich by the Wallraff group [7] and on the Zuchongzhi 2.1 superconducting quantum processor [8]. Both experimental error rates are above the pseudo-threshold for this code relative to a single qubit; see Physics viewpoint for a summary [9]. Magic states have been created on the latter processor [10]. Lattice surgery on the surface-17 code has been realized by splitting the code into two repetition codes by the Wallraff group [11]. The device noise can be used to develop a decoder without relying on a theoretical noise model [12].Notes
Subject of various numerical studies examining the code under noise models and architectures specific to trapped ions [1,13,14] and superconducting circuits [15–17].Cousins
- \([[9,1,3]]\) Shor 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).
- \([[30,8,3]]\) Bring code— Bring’s code and the surface-17 code have been compared numerically [18].
- \([[9,1,4,3]]\) Nine-qubit Bacon-Shor code— The \([[9,1,3]]\) rotated surface code is a CSS gauge fixing of the nine-qubit Bacon-Shor code [2].
Member of code lists
- 2D stabilizer codes
- Lattice qubit stabilizer codes
- Quantum codes
- Quantum codes based on homological products
- Quantum codes with fault-tolerant gadgets
- Quantum codes with notable decoders
- Quantum codes with transversal or permutation-based gates
- Qubit CSS codes
- Realized quantum codes
- Small-distance qubit stabilizer codes and friends
- Surface code and friends
- Topological codes
Primary Hierarchy
Generalized homological-product qubit CSS codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Kitaev surface codeCDSC Twist-defect surface QLDPC Qubit CSS Generalized homological-product Lattice stabilizer Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum
Rotated surface codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Parents
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[9,1,3]]\) Surface-17 code
References
- [1]
- Y. Tomita and K. M. Svore, “Low-distance surface codes under realistic quantum noise”, Physical Review A 90, (2014) arXiv:1404.3747 DOI
- [2]
- A. Cross and D. Vandeth, “Small Binary Stabilizer Subsystem Codes”, (2025) arXiv:2501.17447
- [3]
- Qiskit Community, “Qiskit QEC framework”, URL
- [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]
- 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”, Physical Review X 15, (2025) arXiv:2402.17761 DOI
- [6]
- H. E. Ercan, J. Ghosh, D. Crow, V. N. Premakumar, R. Joynt, M. Friesen, and S. N. Coppersmith, “Measurement-free implementations of small-scale surface codes for quantum-dot qubits”, Physical Review A 97, (2018) arXiv:1708.08683 DOI
- [7]
- S. Krinner et al., “Realizing repeated quantum error correction in a distance-three surface code”, Nature 605, 669 (2022) arXiv:2112.03708 DOI
- [8]
- Y. Zhao et al., “Realization of an Error-Correcting Surface Code with Superconducting Qubits”, Physical Review Letters 129, (2022) arXiv:2112.13505 DOI
- [9]
- L. Frunzio and S. Singh, “Error-Correcting Surface Codes Get Experimental Vetting”, Physics 15, (2022) DOI
- [10]
- Y. Ye et al., “Logical Magic State Preparation with Fidelity Beyond the Distillation Threshold on a Superconducting Quantum Processor”, (2023) arXiv:2305.15972
- [11]
- I. Besedin et al., “Lattice surgery realized on two distance-three repetition codes with superconducting qubits”, Nature Physics (2026) arXiv:2501.04612 DOI
- [12]
- A. Remm et al., “Experimentally Informed Decoding of Stabilizer Codes Based on Syndrome Correlations”, (2025) arXiv:2502.17722
- [13]
- C. J. Trout, M. Li, M. Gutiérrez, Y. Wu, S.-T. Wang, L. Duan, and K. R. Brown, “Simulating the performance of a distance-3 surface code in a linear ion trap”, New Journal of Physics 20, 043038 (2018) arXiv:1710.01378 DOI
- [14]
- D. M. Debroy, M. Li, S. Huang, and K. R. Brown, “Logical Performance of 9 Qubit Compass Codes in Ion Traps with Crosstalk Errors”, (2020) arXiv:1910.08495
- [15]
- R. Versluis, S. Poletto, N. Khammassi, B. Tarasinski, N. Haider, D. J. Michalak, A. Bruno, K. Bertels, and L. DiCarlo, “Scalable Quantum Circuit and Control for a Superconducting Surface Code”, Physical Review Applied 8, (2017) arXiv:1612.08208 DOI
- [16]
- T. E. O’Brien, B. Tarasinski, and L. DiCarlo, “Density-matrix simulation of small surface codes under current and projected experimental noise”, npj Quantum Information 3, (2017) arXiv:1703.04136 DOI
- [17]
- B. M. Varbanov, F. Battistel, B. M. Tarasinski, V. P. Ostroukh, T. E. O’Brien, L. DiCarlo, and B. M. Terhal, “Leakage detection for a transmon-based surface code”, npj Quantum Information 6, (2020) arXiv:2002.07119 DOI
- [18]
- J. Conrad, C. Chamberland, N. P. Breuckmann, and B. M. Terhal, “The small stellated dodecahedron code and friends”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, 20170323 (2018) arXiv:1712.07666 DOI
Page edit log
- Remmy Zen (2024-07-15) — most recent
- Kenneth R. Brown (2022-06-12)
- Victor V. Albert (2022-06-12)
Cite as:
“\([[9,1,3]]\) Surface-17 code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/surface-17