Tricycle code[1,2]

Description

A three-dimensional generalization of 2BGA and bivariate bicycle codes, constructed as a three-block CSS code from three commuting elements \(a,b,c\) of the group algebra \(\mathbb{F}_2[G]\) of a finite Abelian group \(G\); equivalently, the case \(t=3\) of MM codes. The redundancy of the \(Z\)-type checks makes tricycle codes single-shot decodable in the \(Z\) basis, and they host constant-depth transversal \(CCZ\) circuits useful for magic-state generation.

The code is supported on \(n=3|G|\) qubits arranged in three blocks, with check and metacheck matrices \begin{align} H_X = \left[A^T\, B^T\, C^T\right],\quad H_Z = \left[ \begin{array}{ccc} C&0&A\\ 0&C&B\\ B&A&0 \end{array} \right],\quad H_{\text{meta}} = \left[B\, A\, C\right]~, \tag*{(1)}\end{align} where \(A\), \(B\), and \(C\) are Kronecker products of circulant matrices representing \(a\), \(b\), and \(c\), and where \(H_{\text{meta}} H_Z = 0\) [2]. There are \(|G|\) \(X\)-type checks of weight \(\text{wt}(a)+\text{wt}(b)+\text{wt}(c)\) and \(3|G|\) \(Z\)-type checks; codes are often labeled by their element weights, e.g., \(2\)-\(2\)-\(2\) or \(4\)-\(2\)-\(2\). Decomposing \(G\) into one, two, or three cyclic factors yields generalized, bivariate, and trivariate tricycle codes, respectively [2].

Connected tricycle codes, i.e., those for which the supports of \(a\), \(b\), and \(c\) generate \(G\), are three-fold homological products over the ring \(R=\mathbb{F}_2[N]\), where \(N\) is the intersection of the three support subgroups [2]. Example codes include \([[48,6,(8,4)]]\), \([[108,15,(12,6)]]\), and \([[270,24,(15,8)]]\) codes, with distances listed as \((d_X,d_Z)\) [2], as well as a \([[72,6,(12,6)]]\) code [1].