Binary dihedral permutation-invariant code[1] 

Description

Multi-qubit code designed to realize gates from the binary dihedral group transversally. Can also be interpreted as a single-spin code. The codespace projection is a projection onto an irrep of the binary dihedral group \( \mathsf{BD}_{2N} = \{\omega I, X, P\} \) of order \(8N\), where \( \omega \) is a \( 2N \)th root of unity, and \( P = \text{diag} ( 1, \omega^2) \).

The construction includes three families. The first family has parameters \(((2m+3,2,3))\) for \(m\) not a power of two, realizing binary dihedral transversal gates that are not possible to realize in any qubit stabilizer code [1; Prop. 1]. The second family is the case of \(m\) being a power of two, corresponding to \(((2^{m-1}+3,2,3))\) codes, each realizing a member of the Clifford hierarchy transversally. The third family consists of \(((n,2,d))\) codes with \(n = \frac{1}{4}(3d^2+6d-7+2(d\text{ mod }8))\), realizing \(S\) and \(T\) gates transversally.

Transversal Gates

Binary dihedral group gates can be realized transversally, which include subgroups of any level of the Clifford hierarchy and subgroups which cannot be realized by any qubit stabilizer code.

Parents

Cousins

  • Small-distance block quantum code — The first and second families of binary dihedral permutation-invariant codes have distance three, and the third family has the member \(((27,2,5))\).
  • Combinatorial permutation-invariant code — The \(Q_{3,1,2m-4,+}\) and \(Q_{3,1,2^m-4,+}\) combinatorial permutation-invariant codes reduce to the \(((2m+3,2,3))\) and \(((2^{m-1}+3,2,3))\) binary dihedral permutation-invariant codes, respectively [2; Prop. 5.6].
  • XP stabilizer code — Binary dihedral permutation invariant codewords form error spaces of XP stabilizer codes.
  • \([[2^r-1, 1, 3]]\) simplex 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.
  • \([[49,1,5]]\) triorthogonal code — The \(((27,2,5))\) binary dihedral permutation-invariant code realizes the \(T\) gate transversally, but requires fewer qubits than the \([[49,1,5]]\) triorthogonal code.
  • \([[15,1,3]]\) quantum Reed-Muller 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. Kubischta and I. Teixeira, “The Not-So-Secret Fourth Parameter of Quantum Codes”, (2023) arXiv:2310.17652
[2]
A. Aydin, M. A. Alekseyev, and A. Barg, “A family of permutationally invariant quantum codes”, (2024) arXiv:2310.05358
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Zoo Code ID: binary_dihedral_permutation_invariant

Cite as:
“Binary dihedral permutation-invariant code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/binary_dihedral_permutation_invariant
BibTeX:
@incollection{eczoo_binary_dihedral_permutation_invariant, title={Binary dihedral permutation-invariant code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/binary_dihedral_permutation_invariant} }
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“Binary dihedral permutation-invariant code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/binary_dihedral_permutation_invariant

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/permutation_invariant/binary_dihedral_permutation_invariant.yml.