Permutationally self-dual (PSD) CSS code[1]

Cousins

• Dual linear code— The \(X\)- and \(Z\)-type stabilizer spaces of a permutationally self-dual CSS code are permutation-equivalent binary linear codes.
• \([[4,2,2]]\) Four-qubit code— Concatenating a PSD code with copies of the \([[4,2,2]]\) code along a fixed-point-free involutory \(XZ\)-duality yields a self-dual CSS code [4; Thm. 3.5].
• Concatenated qubit code— For a suitable concatenation map, every level of the recursive concatenation of an \([[n_0,1,d_0]]\) PSD seed code is finite-size Kramers-Wannier self-dual under the statistical mechanical mapping of bit-flip decoding [5; Lemma 1]. If the resulting zero-rate family has a unique threshold, this self-duality constrains its optimal bit-flip threshold [5].
• Bipartite cyclic cluster (BCC) code— BCC codes with \(k=2\) are permutationally self-dual. Their \(XZ\)-duality interchanges qubits \((m,1)\) and \((-m,2)\) [7].
• Binarized-and-concatenated (B&C) phantom code— B&C phantom codes built from the CSS quantum QR codes over \(\mathbb{F}_4\) of prime length \(p\equiv 3 \bmod 8\), which have \(H_X=H_Z\), are permutationally self-dual. Transversal Hadamard composed with the block permutation induced by the coordinate map \(i\mapsto 2i \bmod p\) preserves the stabilizer group [8]. B&C phantom codes built from Hermitian self-dual starting codes are self-dual CSS codes, and hence are also permutationally self-dual [8].
• Haah cubic code (CC)— CSS cubic codes are qubit Abelian 2BGA codes [9; Sec. III.A], so they are permutationally self-dual [10; Thm. 6]. For cubic code 1, transversal Hadamard composed with the spatial inversion \(\vec{r}\to-\vec{r}\) and a swap of the two qubits on each site exchanges the \(X\)- and \(Z\)-type stabilizers [3].
• Group-action lift with active orthogonality (GALA) code— GALA codes satisfying the block-index involution condition have check matrices related by a qubit permutation, \(H_Z = H_X \tau\) [11]. Such a code is permutationally self-dual whenever \(\tau\) is an involution, since \(\tau\) then also maps \(H_Z\) onto \(H_X\).
• Hypergraph product (HGP) code— Square hypergraph product codes are permutationally self-dual. Reflecting each of the two grids of qubits about its principal diagonal is an involutive \(XZ\)-duality. Composed with transversal Hadamard, it yields the Hadamard-SWAP logical gate [12; Sec. 3.1].
• Rotated surface code— The \([[L^2,1,L]]\) rotated surface code is permutationally self-dual for odd \(L\), with an \(XZ\)-duality given by a \(\pi/2\) rotation of the qubit grid [13]. For even \(L\), the numbers of independent \(X\)- and \(Z\)-type stabilizer generators differ because they sum to the odd number \(L^2-1\). No permutation can then exchange the two stabilizer groups.
• Kitaev surface code— Surface codes on self-dual cellulations of closed surfaces, such as the toric code, are permutationally self-dual. The isomorphism between the cellulation and its dual induces an edge permutation that exchanges vertex and face stabilizers [1; Sec. 2.3]. The planar code is also permutationally self-dual, with an \(XZ\)-duality given by a reflection along a diagonal [1; Fig. 1].
• Abelian two-block group-algebra code— Qubit Abelian 2BGA codes are permutationally self-dual [10; Thm. 6]. Exchanging the two blocks of qubits while inverting each group element is an involutive \(XZ\)-duality [14; Sec. 3].