Directional tile code[1]

Description

Member of a family of tile codes whose stabilizer supports form an ordered connected string on a planar square lattice, chosen so that the same string is a nearest-neighbor walk measuring the stabilizer generator it defines. One word in the four lattice directions fixes the \(X\)-type tile, its dual-lattice \(Z\)-type partner, the measurement schedule, and the iSWAP-based circuit that executes it. These codes combine open planar boundaries, finite-size parameters improving on those of the rotated surface code, and syndrome extraction using only nearest-neighbor gates on a square grid.

A directional word \(\mathfrak{D}=\vec{d}_1\vec{d}_2\cdots\vec{d}_w\) is an ordered string of steps in the four lattice directions. Its trace on the square lattice is a bounded connected string whose edges form the support of an \(X\)-type tile. The partner \(Z\)-type tile carries the same string on the dual lattice, which makes the pair satisfy the commutation condition of the tile-code construction. One further condition is imposed so that the syndrome-extraction circuit is deterministic. Every displacement vector between two edges of the ordered string that has odd vertical displacement must occur an even number of times [1].

A code is obtained by tessellating the directional tile pair over an \(M\times N\) grid of anchor vertices, with both tile types placed at each bulk anchor. Along the boundary, \(X\)- and \(Z\)-type tiles are placed separately. Edges meeting tiles of at most one type are then pruned, along with any tile left with empty support. The surviving edges are the data qubits and the surviving anchors are the check qubits. Embedding the result in a square-grid layout can leave gaps between a check qubit and the data qubits it must reach, and these are filled with routing qubits that keep every step of the walk nearest-neighbor.

One syndrome-extraction round applies one CXSWAP layer per letter of the word, giving depth \(w\), or \(w+2\) including check-qubit preparation and measurement, independently of the code size. The CXSWAP gate is local-Clifford equivalent to the iSWAP gate, and its CNOT part accumulates the stabilizer eigenvalue while its SWAP part advances the walk. A routing step is the same gate applied to a routing qubit prepared in \(|0\rangle\), on which CXSWAP acts as a SWAP. Consecutive rounds alternate between the word and its inverse, which restores the layout without long-range operations.

Representative instances are the \([[60,4,5]]\) and \([[180,4,9]]\) codes from the weight-7 word \(N^2ESEN^2\), the \([[217,10,7]]\) and \([[351,10,9]]\) codes from the weight-9 word \(N^2E^2SE^2N^2\), the \([[182,14,10]]\) and \([[323,14,15]]\) codes from the weight-11 word \(N^2E^2SESE^2N^2\), and the \([[248,20,11]]\) code from the weight-13 word \(N^2E^2SE^3SE^2N^2\) [1]. The \([[323,14,15]]\) code has code-parameter efficiency \(kd^2/n=9.75\), nearly an order of magnitude above that of the rotated surface code. Choosing an unbalanced aspect ratio of the anchor grid yields instances with strongly unbalanced \(X\)- and \(Z\)-type distances, such as a \([[611,20,d_X,d_Z]]\) code with \(d_X\leq 84\) and \(d_Z=7\).