Binary dihedral PI code[1]

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 (strongly) 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 (strongly) transversally, but requires fewer qubits than the \([[49,1,5]]\) triorthogonal 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.
• Subset-Sum-Linear-Programming (SS-LP) code— SS-LP codes are optimized to admit diagonal gates transversally and include \(((7,2,3))\) codes that realize the \(\mathsf{BD}_{16}\) and \(\mathsf{BD}_{32}\) groups transversally, yielding \(T\) and \(\sqrt{T}\) gates, respectively. Larger codes include an \(((8,2,3))\) code that transversally realizes \(\mathsf{BD}_{64}\).