# 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

## Parents

- Permutation-invariant qubit code
- Clifford spin code — Binary dihedral permutation-invariant codes can be interpreted as Clifford single-spin codes.

## 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

## Page edit log

- Victor V. Albert (2023-11-20) — most recent

## 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