Binary dihedral PI code

Binary dihedral PI 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 \( \mathsf{BD}_{2N} = \langle\omega I, X, P\rangle \) of order \(8N\), where \( \omega \) is a \( 2N \)th root of unity, and \( P = \text{diag} ( 1, \omega^2) \).

The construction includes three families and a handful of particular codes. 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. The handful of codes have distance 5 (7, 9, 11, 13) and encode in 27 (49, 73, 107, 147) qubits, all realizing transversal \(T\) gates.

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

PI qubit code
Clifford spin code — Binary dihedral PI codes can be interpreted as Clifford single-spin codes.

Cousins

Small-distance block quantum code — The first and second families of binary dihedral PI codes have distance three, and the third family has the member \(((27,2,5))\).
Combinatorial PI code — The \(Q_{3,1,2m-4,+}\) and \(Q_{3,1,2^m-4,+}\) combinatorial PI codes reduce to the \(((2m+3,2,3))\) and \(((2^{m-1}+3,2,3))\) binary dihedral PI 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 PI 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 PI 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 PI code realizes the \(T\) gate transversally, but requires fewer qubits than the \([[15,1,3]]\) quantum Reed-Muller code.
Triorthogonal code — There exist binary dihedral PI codes that have distance 5 (7, 9, 11, 13) and encode in 27 (49, 73, 107, 147) qubits, all realizing transversal \(T\) gates.

References

[1]: E. Kubischta and I. Teixeira, “Permutation-Invariant Quantum Codes with Transversal Generalized Phase Gates”, (2024) arXiv:2310.17652
[2]: A. Aydin, M. A. Alekseyev, and A. Barg, “A family of permutationally invariant quantum codes”, Quantum 8, 1321 (2024) arXiv:2310.05358 DOI

Page edit log

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

Cite as:

“Binary dihedral PI 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.