\([[48,3,3]]\) transversal \(CCZ\) code[1]

Description

A \([[48,3,3]]\) generalized triorthogonal CSS code whose logical \(CCZ\) gate is a pattern of physical \(T\) and \(T^\dagger\) gates, with no entangling Clifford correction [1]. Its \(X\)-type stabilizer columns are the 48 points of \(PG(5,2)\) outside a solid, i.e., a three-dimensional projective subspace \(PG(3,2)\), which makes the stabilizer space triply even and guarantees \(d_Z\geq3\) for any logical rows [1; Sec. 2]. It is the smallest known such code with three logical qubits and \(d_Z\geq3\) [1].

The stabilizer generator matrix spans a two-weight \([48,6]\) code, called FG48, with 60 codewords of weight 24 and three of weight 32 [1; Prop. 2.3]. FG48 is the unique projective triply-even \([48,6]\) code up to coordinate permutations [1; Prop. 2.3]. The three \(X\)-type logical rows are listed in Ref. [1; Table 2].

Writing each column as \((u,w)\) with \(u\in\mathbb{F}_2^4\) and \(w\in\mathbb{F}_2^2\setminus\{0\}\), the qubits receiving \(T^\dagger\) are those where \begin{align} \gamma(u,w)=u_3+w_1(u_1+u_2+u_3+u_1u_3+u_2u_3+u_2u_4)+w_2(u_3+u_4+u_1u_2+u_1u_4+u_3u_4) \tag*{(1)}\end{align} evaluates to one [1; Prop. 3.9].