Prime-qudit triorthogonal code[1]

Description

Prime-qudit CSS code whose \(X\)-type logical and stabilizer generators satisfy overlap conditions under which a transversal diagonal non-Clifford gate, followed by a diagonal Clifford correction, acts as a diagonal non-Clifford gate on the logical qudits. Cubic overlaps of rows, repeated rows included and possibly with coordinate weights, vanish modulo \(p\) whenever a stabilizer row is involved, and the logical-row overlaps determine the logical gate. The qubit case \(p=2\) consists of the triorthogonal codes and their generalized versions.

In the strict definition, an \(m \times n\) matrix over \(\mathbb{F}_p=\mathbb{Z}_p\) is triorthogonal if its rows \(r_1, \ldots, r_m\) satisfy \(|r_i \cdot r_j| = 0\) and \(|r_i \cdot r_j \cdot r_k| = 0\) modulo \(p\), where addition and multiplication are done on \(\mathbb{F}_p\). The triorthogonal prime-qudit CSS code associated with the matrix is constructed by mapping nonzero entries in self-orthogonal rows to \(X\) operators, and \(Z\) operators for each row in the orthogonal complement [1,2]. A variant for odd \(p\) drops the pairwise condition and allows nonzero coordinate weights [3]. It requires the weighted cubic overlap to vanish on every triple of rows except a logical row repeated three times, where it must be nonzero [3].