Recursive 3D self-correcting code [1]

Description

Family of geometrically local 3D CSS codes, each encoding one logical qubit, whose stabilizer Hamiltonians are self-correcting [1]. Each iteration replaces every square of the Tanner square complex by two parallel layers joined by a thickness-two strip, doubling the syndrome weight of one error type [1; Sec. 3.2]. Alternating \(X\)-type and \(Z\)-type iterations makes this doubling occur at all length scales and for both error types.

The Tanner square complex of a code is its Tanner graph with squares \((x,q,z,q')\) attached for pairs of qubits shared by checks \(x\) and \(z\) [2]. The recursion starts from the \([[5,1,2]]\) rotated surface code [1; Sec. 3.3]. Before the replacement, an iteration scales the embedding of this complex in \(\mathbb{R}^3\) by a large constant and randomly perturbs its vertices to lower the density of squares. The replacing complexes, \(U_X\) in the \(X\)-iteration and \(U_Z\) in the \(Z\)-iteration, are glued along the boundaries of neighboring squares. The complex \(U_X\) resembles a piece of a thickness-one 3D surface code slab whose front and back boundaries condense \(X\)-type membranes, and \(U_Z\) is its \(Z\)-type counterpart [1; Remark 3.7]. Stabilizer generators have weight at most seven, and each qubit lies in at most seven generators [1; Lemma 3.14 and Thm. 3.16].