\([[2^r-1, 1, 3]]\) quantum Reed-Muller code[1,2] 

Description

Member of CSS code family constructed with a first-order punctured RM\((1,r)\) \([2^r-1,r+1,2^{r-1}-1]\) code and its even subcode for \(r \geq 3\). Each code transversally implements a member of an infinite family of diagonal gates from the Clifford hierarchy [3].

Transversal Gates

\(Z\)-rotation by angle \(-\pi/2^{r-1}\) [2]. These are the smallest qubit stabilizer codes with such a gate [4].

Parents

Children

References

[1]
D. Gottesman and I. L. Chuang, “Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations”, Nature 402, 390 (1999) arXiv:quant-ph/9908010 DOI
[2]
B. Zeng et al., “Local unitary versus local Clifford equivalence of stabilizer and graph states”, Physical Review A 75, (2007) arXiv:quant-ph/0611214 DOI
[3]
S. X. Cui, D. Gottesman, and A. Krishna, “Diagonal gates in the Clifford hierarchy”, Physical Review A 95, (2017) arXiv:1608.06596 DOI
[4]
S. Koutsioumpas, D. Banfield, and A. Kay, “The Smallest Code with Transversal T”, (2022) arXiv:2210.14066
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Zoo Code ID: diagonal_clifford

Cite as:
“\([[2^r-1, 1, 3]]\) quantum Reed-Muller code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/diagonal_clifford
BibTeX:
@incollection{eczoo_diagonal_clifford,
  title={\([[2^r-1, 1, 3]]\) quantum Reed-Muller code},
  booktitle={The Error Correction Zoo},
  year={2022},
  editor={Albert, Victor V. and Faist, Philippe},
  url={https://errorcorrectionzoo.org/c/diagonal_clifford}
}
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Permanent link:
https://errorcorrectionzoo.org/c/diagonal_clifford

Cite as:

“\([[2^r-1, 1, 3]]\) quantum Reed-Muller code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/diagonal_clifford

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/rm/diagonal_clifford.yml.