## 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-eight \(X\)-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.

The code can also be thought of as a code on all corners of a tesseract except one [4].

## Magic

## Encoding

## Transversal Gates

## Gates

## Fault Tolerance

## Threshold

## Parents

- Tetrahedral color code — The \([[15,1,3]]\) code is a tetrahedral color code defined on a single tetrahedron.
- \([[2^r-1,1,3]]\) simplex code
- XS stabilizer code — The \([[15,1,3]]\) code can be viewed as an XS stabilizer code [14; Exam. 6.4].
- Triorthogonal code — The \([[15, 1, 3]]\) code is a triorthogonal code [15].

## Cousins

- Doubled color code — The \([[15,1,3]]\) code can be viewed as a (gauge-fixed) doubled color code obtained from the Steane code via the doubling transformation [16].
- \([[7,1,3]]\) Steane code — The \([[15,1,3]]\) code can be viewed as a (gauge-fixed) doubled color code obtained from the Steane code via the doubling transformation [16]. A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [8,10–12,17]. The \([[105,1]]\) concatenation of the \([[15,1,3]]\) and Steane codes allows for a universal gate set consisting of gates that are transversal w.r.t. to two different partitions [18,19].
- Concatenated Steane code — The \([[105,1]]\) concatenation of the \([[15,1,3]]\) and Steane codes allows for a universal gate set consisting of gates that are transversal w.r.t. to two different partitions [18,19].
- Concatenated qubit code — The concatenated \([[15,1,3]]\) code has a measurement threshold less than one [20].
- Tensor-network code — The quantum Lego framework yields an \([[8,1,2]]\) stabilizer code admits a transversal logical \(T\) gate that originates from that of a trivial (distance-one) \([[7,1]]\) code. This code, in turn, is obtained from the \([[15,1,3]]\) code [21].
- Binary dihedral PI code — The \(((11,2,3))\) binary dihedral PI code realizes the \(T\) gate transversally, but requires fewer qubits than the \([[15,1,3]]\) quantum Reed-Muller code.
- \([[10,1,2]]\) CSS code — The \([[10,1,2]]\) code can be obtained by morphing the \([[15,1,3]]\) code [22].
- \([[8,3,2]]\) CSS code — The \([[8,3,2]]\) code can be obtained from a subset of physical qubits of the \([[15,1,3]]\) code [22].
- \([[15, 7, 3]]\) quantum Hamming code — Gauging out six of the seven logical qubits of the \([[15,7,3]]\) code yields the \([[15,1,3]]\) code [23].

## References

- [1]
- E. Knill, R. Laflamme, and W. Zurek, “Threshold Accuracy for Quantum Computation”, (1996) arXiv:quant-ph/9610011
- [2]
- A. Steane, “Quantum Reed-Muller Codes”, (1996) arXiv:quant-ph/9608026
- [3]
- 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
- [4]
- R. Raussendorf, “Key ideas in quantum error correction”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370, 4541 (2012) 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]
- J. Haah et al., “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI
- [7]
- R. Zen et al., “Quantum Circuit Discovery for Fault-Tolerant Logical State Preparation with Reinforcement Learning”, (2024) arXiv:2402.17761
- [8]
- 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
- [9]
- S. Koutsioumpas, D. Banfield, and A. Kay, “The Smallest Code with Transversal T”, (2022) arXiv:2210.14066
- [10]
- 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
- [11]
- 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
- [12]
- D. Banfield and A. Kay, “Implementing Logical Operators using Code Rewiring”, (2023) arXiv:2210.14074
- [13]
- A. W. Cross, D. P. DiVincenzo, and B. M. Terhal, “A comparative code study for quantum fault-tolerance”, (2009) arXiv:0711.1556
- [14]
- M. A. Webster, B. J. Brown, and S. D. Bartlett, “The XP Stabiliser Formalism: a Generalisation of the Pauli Stabiliser Formalism with Arbitrary Phases”, Quantum 6, 815 (2022) arXiv:2203.00103 DOI
- [15]
- S. Nezami and J. Haah, “Classification of small triorthogonal codes”, Physical Review A 106, (2022) arXiv:2107.09684 DOI
- [16]
- S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
- [17]
- F. Butt et al., “Fault-Tolerant Code-Switching Protocols for Near-Term Quantum Processors”, PRX Quantum 5, (2024) arXiv:2306.17686 DOI
- [18]
- T. Jochym-O’Connor and R. Laflamme, “Using Concatenated Quantum Codes for Universal Fault-Tolerant Quantum Gates”, Physical Review Letters 112, (2014) arXiv:1309.3310 DOI
- [19]
- T. Jochym-O’Connor, A. Kubica, and T. J. Yoder, “Disjointness of Stabilizer Codes and Limitations on Fault-Tolerant Logical Gates”, Physical Review X 8, (2018) arXiv:1710.07256 DOI
- [20]
- D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
- [21]
- R. Shen, Y. Wang, and C. Cao, “Quantum Lego and XP Stabilizer Codes”, (2023) arXiv:2310.19538
- [22]
- M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
- [23]
- A. Kubica and M. E. Beverland, “Universal transversal gates with color codes: A simplified approach”, Physical Review A 91, (2015) arXiv:1410.0069 DOI

## Page edit log

- Remmy Zen (2024-07-15) — most recent
- Balint Pato (2023-03-30)
- 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.), 2024. https://errorcorrectionzoo.org/c/stab_15_1_3