# \([[15,1,3]]\) Reed-Muller code

## Description

\([[15,1,3]]\) 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

Magic-state distillation scaling exponent \( \gamma= \log_d (n/k)\approx 2.46\) [1].

## Transversal Gates

A transversal logical \(T^\dagger\) is implemented by applying a \(T\) gate on every qubit [2][3][4].

## Gates

Code is often used in magic-state distillation protocols because of its transversal \(T\) gate [5].

## Fault Tolerance

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 [3][6].

## Notes

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" [7].

## Parents

- Quantum Reed-Muller code — The \([[15,1,3]]\) code is often noted as the 15-qubit quantum Reed-Muller code in the literature.
- Triorthogonal code — The \([[15, 1, 3]]\) code is a triorthogonal code [8]
- Color code — The \([[15,1,3]]\) code is a 3D color code.

## Zoo code information

## References

- [1]
- J. Haah et al., “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017). DOI; 1703.07847
- [2]
- E. Knill, R. Laflamme, and W. Zurek, “Threshold Accuracy for Quantum Computation”. quant-ph/9610011
- [3]
- J. T. Anderson, G. Duclos-Cianci, and D. Poulin, “Fault-Tolerant Conversion between the Steane and Reed-Muller Quantum Codes”, Physical Review Letters 113, (2014). DOI; 1403.2734
- [4]
- E. T. Campbell, B. M. Terhal, and C. Vuillot, “Roads towards fault-tolerant universal quantum computation”, Nature 549, 172 (2017). DOI; 1612.07330
- [5]
- S. Bravyi and A. Kitaev, “Universal quantum computation with ideal Clifford gates and noisy ancillas”, Physical Review A 71, (2005). DOI; quant-ph/0403025
- [6]
- 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). DOI; 1703.03860
- [7]
- Michael Vasmer and Aleksander Kubica, “Morphing quantum codes”. 2112.01446
- [8]
- Sepehr Nezami and Jeongwan Haah, “Classification of Small Triorthogonal Codes”. 2107.09684

## Cite as:

“\([[15,1,3]]\) Reed-Muller code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stab_15_1_3