# Surface-17 code[1]

## Description

A \([[9,1,3]]\) rotated surface code named for the sum of its 9 data qubits and 8 syndrome qubits. It uses the smallest number of qubits to perform fault-tolerant error correction on a surface code with parallel syndrome extraction.

## 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 [2].Fault-tolerant logical zero and logical plus state preparation in all-to-all and 2D grid connectivity with flag qubits [3].

## Transversal Gates

Pauli gates, CNOT gate, and \(H\) gate (with relabeling).

## Decoding

Lookup table [1].

## Fault Tolerance

Measurement-free fault-tolerant logical zero state preparation in nearest-neighbor qubit connectivity [2].Fault-tolerant logical zero and logical plus state preparation in all-to-all and 2D grid connectivity with flag qubits [3].

## Realizations

Implemented at ETH Zurich by the Wallraff group [4] and on the Zuchongzhi 2.1 superconducting quantum processor [5]. Both experimental error rates are above the pseudo-threshold for this code relative to a single qubit; see Physics viewpoint for a summary [6]. Magic state have been created on the latter processor [7].

## Notes

Subject of various numerical studies examining the system for noises and architectures specific to trapped ions [1,8,9] and superconducting circuits [10–12]

## Parents

## 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 numerically [13].

## 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]
- 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
- [3]
- R. Zen et al., “Quantum Circuit Discovery for Fault-Tolerant Logical State Preparation with Reinforcement Learning”, (2024) arXiv:2402.17761
- [4]
- S. Krinner et al., “Realizing repeated quantum error correction in a distance-three surface code”, Nature 605, 669 (2022) arXiv:2112.03708 DOI
- [5]
- Y. Zhao et al., “Realization of an Error-Correcting Surface Code with Superconducting Qubits”, Physical Review Letters 129, (2022) arXiv:2112.13505 DOI
- [6]
- L. Frunzio and S. Singh, “Error-Correcting Surface Codes Get Experimental Vetting”, Physics 15, (2022) DOI
- [7]
- Y. Ye et al., “Logical Magic State Preparation with Fidelity Beyond the Distillation Threshold on a Superconducting Quantum Processor”, (2023) arXiv:2305.15972
- [8]
- C. J. Trout et al., “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
- [9]
- D. M. Debroy et al., “Logical Performance of 9 Qubit Compass Codes in Ion Traps with Crosstalk Errors”, (2020) arXiv:1910.08495
- [10]
- R. Versluis et al., “Scalable Quantum Circuit and Control for a Superconducting Surface Code”, Physical Review Applied 8, (2017) arXiv:1612.08208 DOI
- [11]
- 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
- [12]
- B. M. Varbanov et al., “Leakage detection for a transmon-based surface code”, npj Quantum Information 6, (2020) arXiv:2002.07119 DOI
- [13]
- J. Conrad et al., “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:

“Surface-17 code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/surface-17