Two-branch coset CSS code[1]

Description

Qubit CSS code whose base stabilizer generator matrices place the support of each check on translated cosets of a multiplicative subgroup \(M\) of a finite field, once in each of two copies of the qubit set called branches [1]. The branch coefficients are chosen so that any overlap between an \(X\)-type and a \(Z\)-type check occurs once in each branch and is therefore even. Any overlap between two checks of the same type occurs in at most one branch, which excludes four-cycles. A cyclic lift of the base pair by circulant permutation matrices can then randomize the edge connections while preserving these constraints [1].

Fix a column weight \(J\), a finite field \(\mathbb{F}\), and a subgroup \(M\) of \(\mathbb{F}^{\times}\) of order \(L/2\). Qubits are indexed by triples \((\lambda,t,h)\) with branch \(\lambda\in\{0,1\}\), translation \(t\in\mathbb{F}\), and \(h\in M\), and checks of each type by pairs \((i,r)\) with \(i<J\) and \(r\in\mathbb{F}\). Each branch carries coefficient vectors \(a^{(\lambda)},b^{(\lambda)}\in\mathbb{F}^J\), and qubit \((\lambda,t,h)\) lies in the \(X\)-type checks \((i,t+a^{(\lambda)}_i h)\) and the \(Z\)-type checks \((j,t+b^{(\lambda)}_j h)\) for all \(i,j<J\). Every check then has weight \(L=2|M|\), every qubit has degree \(J\) in each matrix, and the base length is \(2|\mathbb{F}||M|\). Under the nonzero-difference conditions of Ref. [1; Thm. 2], an \(X\)-type check \((i,r)\) and a \(Z\)-type check \((j,s)\) share a qubit in branch \(\lambda\) exactly when \(s-r\in(b^{(\lambda)}_j-a^{(\lambda)}_i)M\), and then exactly one. The coset equalities \((b^{(0)}_j-a^{(0)}_i)M=(b^{(1)}_j-a^{(1)}_i)M\) for all \(i,j\) therefore make the two matrices orthogonal [1; Thm. 2]. Disjointness of the same-type difference cosets \((a^{(0)}_{i'}-a^{(0)}_i)M\) and \((a^{(1)}_{i'}-a^{(1)}_i)M\), and likewise for \(b\), excludes same-type four-cycles [1; Thm. 3]. These coset conditions turn the search for a regular base pair into a finite check on field coefficients. When every \(X\)-type and \(Z\)-type check share zero or two base qubits, the base pair can be given a \(G\)-lift over a cyclic group by circulant permutation matrices. The shifts on each shared pair are constrained so that the two lifted blocks cancel, which preserves orthogonality [1; Thm. 4]. The remaining circulant shifts are chosen to raise the girth and to remove targeted low-weight logical operators [1].

A \(64\)-fold lift of the \((3,10)\)-regular \([[160,76,4]]\) base over \(\mathbb{F}_{16}\) yields a \([[10240,4108,18 \leq d \leq 32]]\) code of rate \(0.401\) whose same-type Tanner graphs have girth at least eight [1]. See Ref. [1; Table 1] for base codes at other column and row weights.