\([[2^D,D,2]]\) hypercube code[1,2][3; Exam. 3] 

Also known as Hyperoctahedron code, Hyperoctahedron color code.

Description

Member of a family of codes defined by placing qubits on a \(D\)-dimensional hypercube, \(Z\)-stabilizers on all two-dimensional faces, and an \(X\)-stabilizer on all vertices. These codes realize gates at the \((D-1)\)-st level of the Clifford hierarchy.

Transversal Gates

CZ, CCZ, CNOT, and generalized \(CZ\) gates at the \((D-1)\)-st level of the Clifford hierarchy [4; Exam. 6.10].

Parents

  • Ball color code — \([[2^D,D,2]]\) hypercube codes can be thought of as small color codes defined on balls constructed from hyperoctahedra [3; Exam. 3], or on lattices with no bulk qubits and cubic boundaries [1,2].
  • Quantum Reed-Muller code — \([[2^D,D,2]]\) hypercube codes are special cases of the \([[2^m,{m \choose r}, 2^r]]\) quantum Reed-Muller codes for \(m=D\) and \(r=1\) [68][5; Exam. 8].
  • XP stabilizer code — The \(D\)th hypercube code can be viewed as an XP stabilizer code with precision \(N = 2^D\) [4; Exam. 6.10].

Children

Cousin

  • Hypercube code — \([[2^D,D,2]]\) hypercube code qubits are placed on vertices of a \(D\)-cube.

References

[1]
A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
[2]
E. Campbell, “The smallest interesting colour code,” Online available at https://earltcampbell.com/2016/09/26/the-smallest-interesting-colour-code/ (2016), accessed on 2019-12-09.
[3]
M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
[4]
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
[5]
N. Rengaswamy et al., “On Optimality of CSS Codes for Transversal T”, IEEE Journal on Selected Areas in Information Theory 1, 499 (2020) arXiv:1910.09333 DOI
[6]
E. T. Campbell and M. Howard, “Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost”, Physical Review A 95, (2017) arXiv:1606.01904 DOI
[7]
E. T. Campbell and M. Howard, “Unifying Gate Synthesis and Magic State Distillation”, Physical Review Letters 118, (2017) arXiv:1606.01906 DOI
[8]
J. Haah and M. B. Hastings, “Codes and Protocols for DistillingT, controlled-S, and Toffoli Gates”, Quantum 2, 71 (2018) arXiv:1709.02832 DOI
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Zoo Code ID: hypercube_quantum

Cite as:
\([[2^D,D,2]]\) hypercube code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/hypercube_quantum
BibTeX:
@incollection{eczoo_hypercube_quantum, title={\([[2^D,D,2]]\) hypercube code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hypercube_quantum} }
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\([[2^D,D,2]]\) hypercube code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/hypercube_quantum

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/hypercube_quantum.yml.