Directional tile code[1]

Description

Member of a family of tile codes on a planar square grid whose \(X\)- and \(Z\)-type stabilizer generators are supported on one connected string traced by a directional word \(\mathfrak{D}=\vec{d}_1\vec{d}_2\cdots\vec{d}_w\) of steps in the four lattice directions. The same word is the schedule of a nearest-neighbor circuit that measures the generators. Each step of the word applies one layer of CXSWAP gates, and each gate both adds a data qubit’s parity to a check qubit and moves that check qubit one site along the walk [1].

A weight-\(w\) generator is thus measured in depth \(w\), with the check-data connectivity generated by moving the check qubits rather than built into the device. The CXSWAP gate is the iSWAP gate up to single-qubit Clifford gates, and iSWAP is native to superconducting devices, the platform the family is designed for. Data qubits sit on lattice edges, \(X\)-type check qubits on vertices, and \(Z\)-type check qubits on plaquettes. At step \(i\), every check qubit interacts with its neighbor in direction \(\vec{d}_i\), acting as the control of the CX for \(X\)-type checks and as the target for \(Z\)-type checks. After \(w\) steps each check qubit has visited every data qubit of its tile and is measured. Consecutive rounds alternate the word and its inverse, which returns every qubit to its starting position. Where the planar embedding leaves a gap between a check qubit and the next data qubit of its walk, a routing qubit prepared in \(|0\rangle\) fills it. On a routing qubit the CXSWAP gate acts as a plain SWAP. Routing qubits are needed only near the boundary, so their number is of order \(O(\sqrt{n})\) [1].

A code is built by placing the \(X\)- and \(Z\)-tile pair at every vertex of an \(M\times N\) grid of anchors and placing boundary tiles of each type separately. Edges covered by tiles of only one type are then pruned, together with any tile emptied by the pruning. The remaining edges are the data qubits and the remaining anchors are the check qubits. Commutation of the two tiles follows from the tile-code mutual condition. Determinism of the measurement circuit requires one more condition on the word: every displacement vector between two edges of the string with odd vertical displacement must occur an even number of times [1]. Directional tile codes are the open-boundary counterparts of directional codes, which apply the same word on a torus.

The weight-\(11\) word \(N^2E^2SESE^2N^2\) yields a \([[323,14,15]]\) code with \(kd^2/n=9.75\), nearly an order of magnitude above that of the rotated surface code [1]. See Ref. [1; Table II] for codes from words of weight \(7\) to \(13\).