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

## 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 \( \gamma= \log_d (n/k)\approx 2.46\) [2].

## Transversal Gates

This code is the smallest qubit stabilizer code with a transversal gate outside of the Clifford group [3]. A transversal logical \(T^\dagger\) is implemented by applying a \(T\) gate on every qubit [1,4,5]. A subsystem version yields a transversal \(CCZ\) gate [6]. The code fails to have a transversal Hadamard gate; otherwise, it would violate the Eastin-Knill theorem.

## Gates

Code is often used in magic-state distillation protocols because of its transversal \(T\) gate [7].

## 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 [4,6,8,9].

## 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" [10].

## Parents

- Three-dimensional color code — The \([[15,1,3]]\) code is a 3D color code.
- \([[2^r-1, 1, 3]]\) quantum Reed-Muller code
- Triorthogonal code — The \([[15, 1, 3]]\) code is a triorthogonal code [11].

## Cousins

- Doubled color code — The \([[15,1,3]]\) code, when extended to a gauge color code, is the smallest doubled color code.
- Concatenated quantum code — The concatenated \([[15,1,3]]\) code has a measurement threshold less than one [12].
- \([[8,3,2]]\) CSS code — The \([[8,3,2]]\) code can be obtained from a subset of physical qubits of the \([[15,1,3]]\) code [10].
- Binary dihedral permutation-invariant code — The \(((11,2,3))\) binary dihedral permutation-invariant code realizes the \(T\) gate transversally, but requires fewer qubits than the \([[15,1,3]]\) quantum Reed-Muller code.

## References

- [1]
- E. Knill, R. Laflamme, and W. Zurek, “Threshold Accuracy for Quantum Computation”, (1996) arXiv:quant-ph/9610011
- [2]
- J. Haah et al., “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI
- [3]
- S. Koutsioumpas, D. Banfield, and A. Kay, “The Smallest Code with Transversal T”, (2022) arXiv:2210.14066
- [4]
- 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) arXiv:1403.2734 DOI
- [5]
- E. T. Campbell, B. M. Terhal, and C. Vuillot, “Roads towards fault-tolerant universal quantum computation”, Nature 549, 172 (2017) arXiv:1612.07330 DOI
- [6]
- A. Paetznick and B. W. Reichardt, “Universal Fault-Tolerant Quantum Computation with Only Transversal Gates and Error Correction”, Physical Review Letters 111, (2013) arXiv:1304.3709 DOI
- [7]
- S. Bravyi and A. Kitaev, “Universal quantum computation with ideal Clifford gates and noisy ancillas”, Physical Review A 71, (2005) arXiv:quant-ph/0403025 DOI
- [8]
- 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
- [9]
- D. Banfield and A. Kay, “Implementing Logical Operators using Code Rewiring”, (2023) arXiv:2210.14074
- [10]
- M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
- [11]
- S. Nezami and J. Haah, “Classification of small triorthogonal codes”, Physical Review A 106, (2022) arXiv:2107.09684 DOI
- [12]
- D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145

## Page edit log

- Balint Pato (2023-03-30) — most recent
- Victor V. Albert (2021-12-09)
- Qingfeng (Kee) Wang (2021-12-07)

## Cite as:

“\([[15,1,3]]\) quantum Reed-Muller code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/stab_15_1_3