\([[15,1,3]]\) CSS code that is most easily thought of as a tetrahedral 3D color code. This code contains 15 qubits, represented by four vertices, four face centers, six edge centers, and one body center. The tetrahedron is cellulated into four identical polyhedron cells by connecting the body center to all four face centers, where each face center is then connected by three adjacent edge centers. Each colored cell corresponds to a weight-8 \(X\)-check, and each face corresponds to a weight-4 \(Z\)-check. A logical \(Z\) is any weight-3 \(Z\)-string along an edge of the entire tetrahedron. The logical \(X\) is any weight-7 \(X\)-face of the entire tetrahedron.
Magic-state distillation scaling exponent \( \gamma= \log_d (n/k)\approx 2.46\) .
This code is the smallest qubit stabilizer code with a transversal gate outside of the Clifford group . A transversal logical \(T^\dagger\) is implemented by applying a \(T\) gate on every qubit [3–5]. A subsystem version yields a transversal \(CCZ\) gate . The code fails to have a transversal Hadamard gate; otherwise, it would violate the Eastin-Knill theorem.
Code is often used in magic-state distillation protocols because of its transversal \(T\) gate .
Combining the Steane code and the 15-qubit Reed-Muller code through a fault-tolerant conversion can result in a universal transversal gate set that does not need magic state distillation [4,6,8,9].
The \([[15,1,3]]\) code can be converted into the smallest known stabilizer code with a fault-tolerant logical \(T\) gate (i.e., the \([[10,1,2]]\) code) through a "morphing procedure" .
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- S. Koutsioumpas, D. Banfield, and A. Kay, “The Smallest Code with Transversal T”, (2022) arXiv:2210.14066
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- D.-X. Quan et al., “Fault-tolerant conversion between adjacent Reed–Muller quantum codes based on gauge fixing”, Journal of Physics A: Mathematical and Theoretical 51, 115305 (2018) arXiv:1703.03860 DOI
- D. Banfield and A. Kay, “Implementing Logical Operators using Code Rewiring”, (2023) arXiv:2210.14074
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- S. Nezami and J. Haah, “Classification of small triorthogonal codes”, Physical Review A 106, (2022) arXiv:2107.09684 DOI
Page edit log
“\([[15,1,3]]\) quantum Reed-Muller code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/stab_15_1_3