Mitten code[1]

Description

Member of a family of LP codes \(\mathrm{LP}(A,B)\) with one-by-two base matrices \(A=[a_0~a_1]\) and \(B=[b_0~b_1]\) whose entries are weight-three elements of the group algebra \(\mathbb{F}_2[G]\) of a non-Abelian group \(G\) [1]. The \(G\)-lift of this shape yields five blocks of \(|G|\) data qubits and two blocks each of \(X\)- and \(Z\)-type checks, hence check weight nine and encoding rate at least \(20\%\). A non-Abelian \(G\) evades the distance ceiling of Abelian lifts of the same shape, allowing distances of about \(20\) with a few hundred data qubits.

In the canonical form of the base matrices, \begin{align} A = [\,g_1+g_2+g_3~~e+g_4+g_5\,]~,\qquad B = [\,h_1+h_2+h_3~~e+h_4+h_5\,]~, \tag*{(1)}\end{align} \(e\) is the identity and \(g_i,h_i\) are group elements, and the left regular representation of \(a_1\) and the right regular representation of \(b_1\) are required to be full rank [1]. Each row of the block check matrices spans five column blocks, four similar ones (the fingers) carrying one regular action of \(G\) and a distinguished one (the thumb) carrying the other, whence the name. The full-rank condition guarantees a canonical logical basis related by the group action, which underlies the modular surgery toolkit [1].

Instances include a \([[540,108,18]]\) code over \(C_9 \rtimes C_{12}\) [1]. See Ref. [1; Table I] for eight instances from \([[150,30,10]]\) to \([[975,195,\leq 24]]\) and their lift groups.