Mitten code[1]

Description

Member of a family of LP codes of stabilizer generator weight nine and encoding rate at least \(20\%\), built from one-by-two base matrices over the group algebra of a non-Abelian group. The shape of the base matrices fixes the rate, while taking the lift group non-Abelian removes a distance ceiling suffered by Abelian lifts of the same shape, allowing distances of about \(20\) with only a few hundred data qubits. Their small block size, together with a logical basis permuted by the group action, admits low-overhead and highly parallel logical operations.

The code is \(\mathrm{LP}(A,B)\), i.e., the \(G\)-lift of base matrices \(A=[a_0~a_1]\) and \(B=[b_0~b_1]\) over the group algebra \(\mathbb{F}_2[G]\) of a non-Abelian group \(G\), in the canonical form \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} where \(e\) is the identity and \(g_i,h_i\) are group elements, and where the left regular representation of \(a_1\) and the right regular representation of \(b_1\) are required to be full rank. The resulting code has five blocks of \(|G|\) data qubits and two blocks each of \(X\)- and \(Z\)-type checks. The name comes from the block structure of the check matrices, in which each check has four similar columns (the fingers) together with a distinguished one (the thumb).

Instances have been obtained by an automated search [1], including \([[150,30,10]]\) over \(C_5 \times S_3\), \([[200,40,12]]\) over \(C_4 \times D_{10}\) (the dihedral group of order ten), \([[300,60,14]]\) over \(C_{10} \times S_3\), \([[500,100,16]]\) over \(C_5 \rtimes C_{20}\), and \([[540,108,18]]\) over \(C_9 \rtimes C_{12}\), whose distances are exact. Three larger reported instances have parameters \([[630,126,\leq 20]]\), \([[780,156,\leq 22]]\), and \([[975,195,\leq 24]]\).