\([[1152,580,\leq 12]]\) co-designed Kasai code[1]

Description

A Kasai code of rate just above one half, co-designed with reconfigurable neutral-atom hardware, in which every qubit permutation required by syndrome extraction reduces to at most one horizontal shift together with one vertical shift or swap on a three-by-thirty-two grid. It is the smaller of the two instances used to demonstrate that ultra-high-rate quantum LDPC codes can reach logical error rates approaching the teraquop regime while remaining implementable under realistic parallel-control constraints.

The code uses \(J=3\) active block rows, \(L=12\) block columns, and lift size \(P=96\), giving \(n=LP=1152\) qubits at rate \(0.503\), girth at least six, and non-commutativity imposed on the index pairs \((0,3)\) and \((1,2)\) [1]. The twelve affine permutations \(f_i(x)=a_ix+b_i\) and \(g_i(x)=c_ix+d_i\) defining the \(G\)-lift are tabulated in Ref. [1]. Its defining co-design property is that all syndrome-extraction transition permutations commute with a reference affine permutation having three orbits of length \(32\), which is what produces the three-by-thirty-two layout and the at-most-one-horizontal-plus-one-vertical-move schedule.

The same construction yields a \([[2304,1156,\leq 14]]\) instance, along with movement-compatible \([[2304,1156,\leq 16]]\) and \([[4608,2308,\leq 22]]\) codes [1].

This code also belongs to a second family: it is permutation equivalent, as verified numerically, to the GALA code \(\mathrm{GALA}_{12,3}(S_3\times\mathbb{Z}_{32})\), with its \([[2304,1156]]\) sibling likewise permutation equivalent to \(\mathrm{GALA}_{12,3}(S_3\times\mathbb{Z}_2\times\mathbb{Z}_{32})\) [2]. The non-Abelian factor \(S_3\) and Abelian factor \(\mathbb{Z}_{32}\) of that presentation reproduce the reference-permutation orbit structure, recovering the same optimal movement schedule from the group data.