\([[2^r-1, 1, 3]]\) simplex code[13] 

Also known as \([[2^r-1, 1, 3]]\) quantum RM code.

Description

Member of color-code code family constructed with a first-order punctured RM\((1,m=r)\) \([2^r-1,r+1,2^{r-1}-1]\) code and its even subcode for \(r \geq 3\). Each code transversally implements a diagonal gate at the \((r-1)\)st level of the Clifford hierarchy [3,4]. Each code is a color code defined on a simplex in \(r-1\) dimensions [5,6], where qubits are placed on the vertices, edges, and faces as well as on the simplex itself.

Transversal Gates

Each code transversally implements a diagonal gate at the \((r-1)\)st level of the Clifford hierarchy in the form of a \(Z\)-rotation by angle \(-\pi/2^{r-1}\) [3,4]. These are the smallest qubit stabilizer codes with such a (strongly) transversal gate [7].

Parents

  • Quantum Reed-Muller code — \([[2^r-1, 1, 3]]\) simplex codes are special cases of the \([[\sum_{i=w+1}^m \binom{m}{i}, \sum_{i=0}^{w} \binom{m}{i}, \sum_{i=w+1}^{r+1} \binom{r+1}{i}]]\) quantum RM codes for \(w=r=0\) and \(m=r-1\) [8; Thm. 1].
  • Color code — Each \([[2^r-1, 1, 3]]\) simplex code is a color code defined on a simplex in \(r-1\) dimensions [5,6].
  • XP stabilizer code — Each \([[2^r-1, 1, 3]]\) simplex code can be viewed as an XP stabilizer code with precision \(N = 2^{r-2}\) [9; Exam. 6.4].
  • Quantum divisible code — \([[2^r-1, 1, 3]]\) simplex codes come from RM\((1,m=r)\) codes, which are \((r-1)\)-even [10,11], and admit transversal gates at levels of the Clifford hierarchy. Building a tower of generalized divisible codes by starting with the Steane code yields the \([[2^r-1, 1, 3]]\) simplex codes [12; Sec. VI.B].
  • Small-distance block quantum code

Children

Cousins

  • \(k\)-orthogonal code — \([[2^r-1, 1, 3]]\) simplex codes are \((r-1)\)-orthogonal [7; Lemma 2].
  • Simplex spherical code — Each \([[2^r-1, 1, 3]]\) simplex code is a color code whose qubits are placed on the vertices, edges, and faces of an \((r-1)\)-simplex [5,6].
  • Binary dihedral permutation-invariant code — The \(((2^{r-1}+3,2,3))\) family of binary dihedral permutation-invariant codes realizes the same transversal gates as the \([[2^r-1, 1, 3]]\) quantum Reed-Muller codes, but require fewer qubits in almost all cases.

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]
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
[3]
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
[4]
S. X. Cui, D. Gottesman, and A. Krishna, “Diagonal gates in the Clifford hierarchy”, Physical Review A 95, (2017) arXiv:1608.06596 DOI
[5]
H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
[6]
B. J. Brown, N. H. Nickerson, and D. E. Browne, “Fault-tolerant error correction with the gauge color code”, Nature Communications 7, (2016) arXiv:1503.08217 DOI
[7]
S. Koutsioumpas, D. Banfield, and A. Kay, “The Smallest Code with Transversal T”, (2022) arXiv:2210.14066
[8]
M. B. Hastings and J. Haah, “Distillation with Sublogarithmic Overhead”, Physical Review Letters 120, (2018) arXiv:1709.03543 DOI
[9]
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
[10]
R. J. McEliece, “On periodic sequences from GF(q)”, Journal of Combinatorial Theory, Series A 10, 80 (1971) DOI
[11]
R. J. McEliece, “Weight congruences for p-ary cyclic codes”, Discrete Mathematics 3, 177 (1972) DOI
[12]
J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
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Zoo Code ID: diagonal_clifford

Cite as:
\([[2^r-1, 1, 3]]\) simplex code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/diagonal_clifford
BibTeX:
@incollection{eczoo_diagonal_clifford, title={\([[2^r-1, 1, 3]]\) simplex 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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\([[2^r-1, 1, 3]]\) simplex 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/edit/main/codes/quantum/qubits/stabilizer/rm/diagonal_clifford.yml.