Tricycle code[1,2]

Description

A three-block CSS code whose commuting matrices represent three elements \(a,b,c\) of the group algebra \(\mathbb{F}_2[G]\) of a finite Abelian group \(G\). The code is the balanced product over \(\mathbb{F}_2[G]\) of the three classical group-algebra codes defined by \(a\), \(b\), and \(c\), extending the two-block 2BGA construction to three homological dimensions [1,2]. The third dimension provides \(Z\)-type metachecks, which make the code single-shot decodable in the \(Z\) basis, and a cup product, which yields constant-depth transversal \(CCZ\) circuits for magic-state generation [2].

Each element defines a two-term complex \(\mathbb{F}_2[G]\xrightarrow{a}\mathbb{F}_2[G]\), and the product of the three complexes is a four-term complex \(\mathbb{F}_2[G]\to\mathbb{F}_2[G]^3\to\mathbb{F}_2[G]^3\to\mathbb{F}_2[G]\) [2]. The \(n=3|G|\) qubits sit at the second term, the \(X\)-type checks at the first, the \(Z\)-type checks at the third, and the \(Z\)-type metachecks at the fourth, giving 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]. Each cyclic factor in the chosen decomposition of \(G\) contributes one circulant tensor factor [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 \(4\)-\(2\)-\(2\). Bivariate tricycle codes, i.e., those over \(\mathbb{Z}_{\ell_x}\times\mathbb{Z}_{\ell_y}\), can be laid out on a triangular lattice on a 2D torus with three qubits per site [3; Appx. D]. All \(X\)-type checks are then translates of a single check, and so are the \(Z\)-type checks within each of the three rows of \(H_Z\) [3; Appx. D].

Tricycle codes are connected when the supports of \(a\), \(b\), and \(c\) generate \(G\) [2]. A \(4\)-\(2\)-\(2\) example is the \([[48,6,(8,4)]]\) code, with distances listed as \((d_X,d_Z)\) [2]. See Refs. [2; Table 1] and [1; Table I] for further instances.