Weakly divisible CSS code[1][2; Defs. I.4 and I.5]

Description

A weakly \(\Delta\)-divisible CSS code for \(\Delta>1\) is a CSS code whose \(X\)-type stabilizer space obeys a signed divisibility condition on two disjoint sets of qubits. The condition allows coordinates outside those sets and does not constrain the logical \(X\) representatives. Weakly four-divisible and weakly eight-divisible CSS codes are also called \(\mathrm{DE}^*\) and \(\mathrm{TE}^*\) codes, respectively.

More explicitly, the binary space \(C_2\) of \(X\)-type stabilizers is weakly \(\Delta\)-divisible if there are disjoint subsets \(M^+,M^-\subseteq[n]\) such that \begin{align} |u\cap M^+|-|u\cap M^-|\equiv0\pmod\Delta \tag*{(1)}\end{align} for every \(u\in C_2\) [2; Defs. I.4 and I.5]. At least one of \(M^+\) and \(M^-\) is required to be nonempty [1]. Coordinates outside \(M^+\cup M^-\) do not contribute to the congruence. Ordinary \(\Delta\)-divisibility is recovered by taking \(M^+=[n]\) and \(M^-=\emptyset\). Weak \(\Delta\)-divisibility implies weak \(\Delta^{\prime}\)-divisibility for every divisor \(\Delta^{\prime}\) of \(\Delta\).

Coset divisibility

A stronger condition applies to a CSS code defined by binary linear codes \(C_2\subseteq C_1\) and a character vector \(y\) fixing the signs of the \(Z\)-type stabilizers. Coset divisibility requires the codeword weights to be constant modulo \(\Delta\) within each coset of \(C_2\) in \(C_1+y\) [3; Thm. 7][4; Cor. 7].

More explicitly, coset divisibility requires \begin{align} \operatorname{wt}(u+w)\equiv\operatorname{wt}(w)\pmod\Delta \tag*{(2)}\end{align} for every \(u\in C_2\) and \(w\in C_1+y\). Taking \(w=y\) shows that every coset-divisible code is weakly divisible with \(M^-=\operatorname{supp}(y)\) and \(M^+=[n]\setminus M^-\). The converse need not hold because weak divisibility tests only this one element of \(C_1+y\), rather than every logical coset representative. Positive signs correspond to \(y=0\), in which case the trivial coset shows that \(C_2\) is a \(\Delta\)-divisible classical code. A \(\Delta\)-coset-divisible code is also \(\Delta^{\prime}\)-coset-divisible for every divisor \(\Delta^{\prime}\) of \(\Delta\).

For \(\Delta=2^\nu\) and \(y=0\), the above is equivalent to \(2^\nu\) dividing \(\operatorname{wt}(u)\) for every \(u\in C_2\), together with \(2^{\nu-1}\) dividing \(\operatorname{wt}(u*w)\) for every \(u\in C_2\) and \(w\in C_1\) [3; Rem. 13]. Here \(*\) denotes the entrywise product.