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\([[7,1,3]]\) twist-defect surface code[1]

Alternative names: \([[7,1,3]]\) triangle code.

Description

A \([[7,1,3]]\) code (different from the Steane code) that is a small example of a twist-defect surface code.

A stabilizer tableau for the code is given by [2; ID 190] \begin{align} \begin{array}{ccccccc} X & I & I & I & I & I & Z \\ I & X & I & I & I & Z & I \\ I & I & X & I & Z & I & I \\ I & I & Z & X & X & Z & I \\ I & Z & I & Y & I & Y & Z \\ Z & I & I & Z & Z & I & X \end{array}~. \tag*{(1)}\end{align} It is one of sixteen distinct \([[7,1,3]]\) codes [3].

Protection

Fully fault-tolerant depolarizing-noise designs using \(13\) or \(15\) total qubits, including ancillas, have exREC pseudothresholds of order \(10^{-4}\) [1].

Transversal Gates

Admits certain transversal order-three single-qubit Clifford gates (e.g., \(SH\)) [1].

Gates

Within the triangle-code architecture, supports the full logical Clifford group using lattice surgery, 1-bit teleportation, and patch reorientation [1].

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum
Twist-defect surface codeLattice stabilizer Stabilizer Hamiltonian-based QECC Quantum
Stellated surface code
Triangular surface code
Parents
Triangular surface code
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[7,1,3]]\) twist-defect surface code

References

[1]
T. J. Yoder and I. H. Kim, “The surface code with a twist”, Quantum 1, 2 (2017) arXiv:1612.04795 DOI
[2]
Qiskit Community. Qiskit QEC framework. https://github.com/qiskit-community/qiskit-qec
[3]
S. Yu, Q. Chen, and C. H. Oh, “Graphical Quantum Error-Correcting Codes”, (2007) arXiv:0709.1780
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Zoo Code ID: twist_defect_7_1_3

Cite as:
“\([[7,1,3]]\) twist-defect surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/twist_defect_7_1_3
BibTeX:
@incollection{eczoo_twist_defect_7_1_3, title={\([[7,1,3]]\) twist-defect surface code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/twist_defect_7_1_3} }
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Permanent link:
https://errorcorrectionzoo.org/c/twist_defect_7_1_3

Cite as:

“\([[7,1,3]]\) twist-defect surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/twist_defect_7_1_3

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/7/twist_defect_7_1_3.yml.