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\([[17,1,5]]\) 4.8.8 color code[1]

Description

Seventeen-qubit doubly even 2D color code that admits a transversal implementation of the logical Clifford group. The smallest distance-five CSS code has \(n=17\) [2]. It is also a normal self-dual CSS code whose transversal Hadamard acts logically, making it suitable as an inner code for fifth-order magic-state distillation [3].

A stabilizer tableau for the code is [4; ID 672e4f2a4f33a38b84d76b7c] \begin{align} \begin{smallmatrix} X & I & I & I & I & I & I & I & I & X & I & I & X & I & I & X & I \\ I & X & I & I & I & I & I & I & I & X & I & I & X & I & I & I & X \\ I & I & X & I & I & I & I & I & I & I & I & X & I & X & X & I & I \\ I & I & I & X & I & I & I & I & X & X & I & I & X & I & I & I & I \\ I & I & I & I & X & I & I & I & X & I & X & I & X & X & X & X & X \\ I & I & I & I & I & X & I & I & I & I & X & X & I & X & I & I & I \\ I & I & I & I & I & I & X & I & X & X & X & I & I & X & X & X & X \\ I & I & I & I & I & I & I & X & I & I & X & X & I & I & X & I & I \\ Z & I & I & I & I & I & I & I & I & Z & I & I & Z & I & I & Z & I \\ I & Z & I & I & I & I & I & I & I & Z & I & I & Z & I & I & I & Z \\ I & I & Z & I & I & I & I & I & I & I & I & Z & I & Z & Z & I & I \\ I & I & I & Z & I & I & I & I & Z & Z & I & I & Z & I & I & I & I \\ I & I & I & I & Z & I & I & I & Z & I & Z & I & Z & Z & Z & Z & Z \\ I & I & I & I & I & Z & I & I & I & I & Z & Z & I & Z & I & I & I \\ I & I & I & I & I & I & Z & I & Z & Z & Z & I & I & Z & Z & Z & Z \\ I & I & I & I & I & I & I & Z & I & I & Z & Z & I & I & Z & I & I \end{smallmatrix}~. \tag*{(1)}\end{align}

Encoding

Logical zero state preparation using reinforcement learning [5].

Transversal Gates

Transversal implementation of the logical Clifford group.

Gates

As an inner code for magic-state distillation, it yields a 69-to-1 fifth-order routine, or a 49-to-1 pipelined routine when preceded by the Steane code [3; Sec. I.A.2].

Cousin

  • \([[49,1,5]]\) triorthogonal code— The \([[49,1,5]]\) triorthogonal code can be viewed as a (gauge-fixed) doubled color code obtained from the \([[17,1,5]]\) 4.8.8 color code via the doubling transformation [6].

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
Color codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum
2D color codeTwist-defect color QLDPC Qubit Generalized homological-product Lattice stabilizer CSS Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum
Square-octagon (4.8.8) color code
Parents
Square-octagon (4.8.8) color code
The \([[17,1,5]]\) color code can be defined on the 4.8.8 tiling [6].
Quantum divisible codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum
Self-dual CSS codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
The smallest distance-five CSS code has \(n=17\) [2].
\([[17,1,5]]\) 4.8.8 color code

References

[1]
H. Bombin and M. A. Martin-Delgado, “Topological Quantum Distillation”, Physical Review Letters 97, (2006) arXiv:quant-ph/0605138 DOI
[2]
M. F. Ezerman, S. Jitman, S. Ling, and D. V. Pasechnik, “CSS-Like Constructions of Asymmetric Quantum Codes”, IEEE Transactions on Information Theory 59, 6732 (2013) arXiv:1207.6512 DOI
[3]
J. Haah, M. B. Hastings, D. Poulin, and D. Wecker, “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI
[4]
S. Burton. “qecdb.org: Quantum Error Correction Database”, URL: https://qecdb.org/
[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]
S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
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Zoo Code ID: stab_17_1_5

Cite as:
“\([[17,1,5]]\) 4.8.8 color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/stab_17_1_5
BibTeX:
@incollection{eczoo_stab_17_1_5, title={\([[17,1,5]]\) 4.8.8 color code}, booktitle={The Error Correction Zoo}, year={2025}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/stab_17_1_5} }
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https://errorcorrectionzoo.org/c/stab_17_1_5

Cite as:

“\([[17,1,5]]\) 4.8.8 color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/stab_17_1_5

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/17/stab_17_1_5.yml.