Generalized quantum divisible code[1]

Description

A level-\(\nu\) generalized quantum divisible code is a CSS code specified by an \(X\)-type stabilizer generator matrix \(S\), a logical generator matrix \(L\), and an odd-integer vector \(t\) [1; Def. V.1]. The matrix \(S\) is \((\nu,t)\)-null, while \(L\) is \((\nu,t)\)-orthonormal. The vertically stacked matrix \(G=[L;S]\) is \((\nu,t)\)-orthogonal. Such codes admit gates at the \(\nu\)th level of the Clifford hierarchy.

The \((\nu,t)\)-norm of a binary vector \(v\) is \begin{align} \lVert v\rVert_{\nu,t}=\sum_i v_i t_i \pmod {2^\nu}. \tag*{(1)}\end{align} Two vectors \(v,w\) are \((\nu,t)\)-orthogonal if \begin{align} \sum_i v_i t_i w_i \equiv 0 \pmod {2^{\nu-1}}. \tag*{(2)}\end{align} A set of vectors is \((\nu,t)\)-orthogonal if the spans of every two disjoint subsets are \((\nu,t)\)-orthogonal. An orthogonal set is null if every vector in its span has zero norm. It is orthonormal if each row has norm one.

Equivalently, \((\nu,t)\)-orthogonality of \(G\) requires \begin{align} 2^{|A|-1}\sum_i t_i\prod_{a\in A}G_{a i}\equiv 0\pmod {2^\nu} \tag*{(3)}\end{align} for every set \(A\) of at least two and at most \(\nu\) distinct rows [1; Lemma III.2]. The conditions range over the rows of the full matrix \(G\), so they constrain logical \(X\) representatives as well as \(X\)-type stabilizers. At level three, all weighted pair products therefore vanish modulo four and all weighted triple products vanish modulo two.

For positive stabilizer signs and \(t=(1,\ldots,1)\), preservation of the code space by the transversal \(Z\)-rotation by \(\pi/2^{\nu-1}\) requires \begin{align} 2^\nu&\mid \mathrm{wt}(w) &&\text{for all }w\in\operatorname{span}(S),\tag*{(4)}\\ 2^{\nu-1}&\mid \mathrm{wt}(w*z) &&\text{for all }w\in\operatorname{span}(S),\ z\in\operatorname{span}(G), \tag*{(5)}\end{align} where \(\mathrm{wt}\) is the Hamming weight and \(*\) is the entrywise product [2; Rem. 13].

Generalized quantum divisible codes can be level-lifted from \(\nu\) to \(\nu+1\) [1; Thm. IV.1]. This procedure recursively yields towers from a ground code [1; Thm. V.6].