\([[132,30,12]]\) self-dual GALA code[1]

Description

A hyperbolic self-dual GALA code whose Hadamard-type and phase-type fold-transversal Clifford operations are implemented physically by depth-one layers of single-qubit gates.

The code is \(\mathrm{GALA}_{12,5}(e\times C_{11})\): the row weight is \(L=12\), the number of active block rows is \(J=5\), the non-Abelian top is trivial, and the Abelian bottom is the cyclic group \(C_{11}\), giving \(n=12\times1\times11=132\) qubits at rate \(0.227\) [1]. Every entry of the \(G\)-lift is a single group element, so the code is monomial. Its \(ZX\) duality comes from a reflection-type sector involution, which is why the Tanner graph has girth four rather than the girth of at least six targeted elsewhere in the family. The duality permutation is the identity, which is what reduces both fold-transversal operations to depth-one layers of single-qubit gates [1].

Since the duality permutation is the identity, the two fold-transversal operators are \begin{align} H_{\tau}=H^{\otimes 132}~,\qquad S_{\tau}=(S^{\dagger})^{\otimes 132}~, \tag*{(1)}\end{align} each a depth-one layer of single-qubit gates [1]. In an orthonormal logical basis, \(H_{\tau}\) acts as fifteen copies of \((\bar H\otimes\bar H)\overline{\mathrm{SWAP}}\), while \(S_{\tau}\) acts as fifteen disjoint logical controlled-\(Z\) gates together with single-logical-qubit phases on thirteen of the thirty logical qubits [1].