Directional code[1]

Description

Member of a family of BB codes on a torus whose stabilizer generators are each supported on a translate of one bounded connected string of the square lattice. The string is traced out by a fixed word in the four lattice directions, and the same word doubles as the schedule of a syndrome-extraction circuit built only from nearest-neighbor iSWAP gates. The exchange action of those gates carries the check qubits through the lattice, so the connectivity needed to measure a high-weight stabilizer generator is produced during the circuit rather than built into the qubit connectivity graph.

Data and check qubits sit on the two sublattices of a checkerboard coloring of \(\mathbb{Z}^2\), and each check qubit has four data-qubit neighbors. Applying a CXSWAP gate between every check qubit and its data-qubit neighbor in a fixed direction accumulates one stabilizer term on each check qubit. It also translates the check-qubit sublattice one step in that direction and the data-qubit sublattice one step the opposite way, changing which data qubits each check qubit is adjacent to. A directional word \(\mathfrak{D}=\vec{d}_1\vec{d}_2\cdots\vec{d}_w\), with each \(\vec{d}_i\) one of the four lattice directions, therefore specifies a depth-\(w\) circuit measuring one weight-\(w\) stabilizer generator per check qubit. Check qubits carry an \(X\)- or \(Z\)-type label, and CXSWAP or CZSWAP gates are applied accordingly. Both gates are local-Clifford equivalent to the iSWAP gate. Not every pairing of a word with an assignment of check-qubit labels yields a deterministic circuit, and the admissible pairings are characterized in Ref. [1].

A finite code is obtained by wrapping the infinite lattice around a parallelogram spanned by two lattice vectors. Axis-parallel vectors give a regular torus, and other choices give a twisted torus. The families constructed explicitly use the words \(N^{\alpha}E^{\beta}N^{\alpha}\) with \(\alpha\geq 1\) and \(\beta\geq 2\), whose stabilizer generators have weight \(2\alpha+\beta\). For each word there is a rotated family on a diamond-shaped twisted torus with \(n=\alpha\beta L^2/2\), and a rectangular family on a regular torus with \(n=3\alpha\beta L^2/4\), both defined for a size parameter \(L\) divisible by four. A filler family on a regular torus supplies an instance with \(n=24\alpha\beta\). All three satisfy \(k\geq 2(2\alpha-1)(\beta-1)\), and the words \(NE^3N\), \(N^2E^2N^2\), \(N^2E^3N^2\), and \(N^2E^4N^2\) attain this bound with four, six, twelve, and eighteen logical qubits, respectively.

Examples include the \([[48,12,4]]\) and \([[192,12,8]]\) rotated \(N^2E^3N^2\) codes, whose code-parameter efficiency is \(kd^2/n=4\). The rotated \(N^2E^4N^2\) codes \([[64,18,4]]\) and \([[256,18,8]]\) reach \(kd^2/n=4.5\). The unrotated toric code is recovered as the \(NE^2N\) code on a square torus.