Tile code[1,2]

Cousins

• Kitaev surface code— Tile codes generalize the surface code; the unrotated surface code is recovered by choosing the surface-code tiles together with an appropriate layout [2].
• Rotated surface code— The rotated surface code is recovered from the tile-code construction using a rotated layout [2]. The most efficient tile codes outperform the rotated surface code in the efficiency \(kd^2/n\) by factors of up to more than 12 [2].
• Bivariate bicycle (BB) code— Tile codes are open-boundary (planar) analogues of bivariate bicycle (BB) codes, which arise by tiling a torus with stabilizer tiles [2]; tile codes instead have open boundaries and thus retain true \(O(1)\)-locality on a planar 2D lattice [2]. Sharing the same bulk stabilizer tiles does not yield the same code: the planar \([[288,8,12]]\) tile code has the same bulk stabilizers as the \((-2,2)\)-BB code, which requires a \(217\times 217\) torus to attain its maximal logical dimension \(k=16\) and yields a \([[98,6,12]]\) code on a \(7\times 7\) torus [1]. The logical dimension of a BB code is determined by the period of the same cellular automaton that generates tile-code logical operators [3]. It is expected that the QEC cycle developed for bivariate bicycle codes [7] can be adapted to tile codes [2].
• Hypergraph product (HGP) code— Pruned hypergraph-product constructions of planar BB codes are recovered by the tile-code construction using an unrotated bulk-stabilizer layout [2].
• La-cross code— Open-boundary La-cross codes are instances of tile codes whose stabilizers are induced by univariate polynomials [3]. La-cross codes with periodic boundary conditions have stabilizers connecting opposite array boundaries [8] and are not tile codes, since tile codes are defined on a finite planar square-lattice layout with open boundaries [2].
• Multivariate multicycle (MM) code— Tile codes and MM codes with two polynomials in two variables arise from the same three-term Koszul complex with qubits placed at the middle level: MM codes take the complex over the group algebra \(\mathbb{F}_q[x,y]/\langle x^{\ell_1}-1,y^{\ell_2}-1\rangle\), yielding periodic boundaries, while tile codes take the higher global sections \(R^1\Gamma\) of the corresponding Koszul complex of vector bundles on \(\mathbb{P}^1\times\mathbb{P}^1\), yielding open boundaries [3,9]. The tile-code construction extends to \(t\) spatial dimensions using \((\mathbb{P}^1)^{t}\) and \(t\) polynomials in \(t\) variables, yielding codes that are local in \(t\) rather than two dimensions [3; Sec. 5.4]. For example, the four-dimensional instance built from four polynomials in \(w,x,y,z\) with \(L=M=N=P=3\) has parameters \([[486,24,d]]\) with \(10\leq d\leq 15\) [3], the same length and logical dimension as the \([[486,24,12]]\) MM code with \(t=4\) over \(\mathbb{Z}_3^{4}\) [9].
• Abelian topological code— The stabilizer tiles of a tile code have to give rise to topological order in the bulk for the code distance to scale with the layout dimensions [2] and for the logical dimension to be \(k=2D^2\) [3]. This is an extra assumption and not part of the definition: writing the \(X\)-tile as a pair of Laurent polynomials \(f,g\) over \(\mathbb{F}_2[x^{\pm},y^{\pm}]\), bulk topological order holds if and only if \(f\) and \(g\) admit no common non-unit factor [3]. Admissible tiles violating this condition, e.g., \(f=g=1+x+y\) in a \(2\times 2\) box, yield valid tile codes such as \([[32,2,2]]\) and \([[288,2,2]]\) whose distance does not scale. Planar tile codes can be obtained from bivariate bicycle codes by condensing anyons of the underlying Abelian topological order at the boundary [1].