Coset-based quantum LDPC code[1]

Description

A two-block CSS code \(Q_G^H(a,b)\) built from the action of a finite group \(G\) on the left cosets \(G/H\) of a subgroup \(H\), generalizing 2BGA codes to a much larger family of quantum LDPC codes. The length \(n=2[G:H]\) is even, being twice the index of \(H\) in \(G\).

The building blocks are permutation matrices \(\mathbf{L}(g)\) and \(\mathbf{R}(g)\) representing, respectively, the left action of \(G\) and the right action of the normalizer \(N_G(H)\) on the \(m=[G:H]\) cosets. These two actions commute, so for group algebra elements \(a\in\mathbb{F}_q[G]\) and \(b\in\mathbb{F}_q[N_G(H)]\) the matrices \(\mathbf{L}(a)\) and \(\mathbf{R}(b)\) commute and yield a valid CSS code with parity-check matrices \begin{align} H_X=[\mathbf{L}(a)\mid \mathbf{R}(b)]~,\qquad H_Z=[-\mathbf{R}(b)^T\mid \mathbf{L}(a)^T] \tag*{(1)}\end{align} on \(n=2m\) qudits, with stabilizer generators of weight at most \(w_a+w_b\), where \(w_a\) and \(w_b\) are the weights of \(a\) and \(b\) [1].

A computer search over non-abelian groups and their non-normal subgroups yields new weight-six codes \([[48,8,6]]\), \([[96,8,10]]\), and \([[224,12,16]]\), and weight-eight codes \([[84,16,8]]\), \([[112,16,10]]\), \([[128,16,12]]\), and \([[168,16,15]]\) [1].