Modular-qudit CSS code[1–5]

Description

An \(((n,K,d))_q\) modular-qudit stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type Pauli strings. Codes can be defined from two classical codes and/or chain complexes over the ring \(\mathbb{Z}_q\) via an extension of qubit CSS-to-homology correspondence to modular qudits [5]. The homology group of the logical operators has a torsion component because the chain complexes are defined over a ring, which yields codes whose logical dimension is not a power of \(q\).

The stabilizer generator matrix, taking values from \(\mathbb{Z}_q\), is of the form \begin{align} H=\begin{pmatrix}0 & H_{Z}\\ H_{X} & 0 \end{pmatrix} \label{eq:parityq} \tag*{(1)}\end{align} such that the rows of the two blocks must be orthogonal \begin{align} H_X H_Z^T=0~. \label{eq:commQ} \tag*{(2)}\end{align} The above condition guarantees that the \(X\)-stabilizer generators, defined in the modular symplectic representation as rows of \(H_X\), commute with the \(Z\)-stabilizer generators associated with \(H_Z\).

For composite \(q\), such codes need not encode an integer number of qudits, but the relation to homology allows for a general structure theorem [6; Thm. 8.1.1]. For prime \(q=p\), properties reminiscent of qubit CSS codes are restored: encoding is based on two related \(p\)-ary linear codes, an \([n,k_X,d_X]_p \) code \(C_X\) and \([n,k_Z,d_Z]_p \) code \(C_Z\), satisfying \(C_X^\perp \subseteq C_Z\). The resulting CSS code has \(k=k_X+k_Z-n\) logical qubits and distance \(d\geq\min\{d_X,d_Z\}\). Specializing to the case when \(C_Z=[n,k,d]_p\) is dual-containing yields an \([[n,2k-n,\geq d_Z]]_p\) self-dual prime-qudit CSS code with \(C_X = C_Z\). The \(H_X\) (\(H_Z\)) block of \(H\) (1) is the parity-check matrix of the code \(C_X\) (\(C_Z\)). The requirement \(C_X^\perp \subseteq C_Z\) guarantees (2). Basis states for the code are, for coset representatives \(\gamma \in C_X/C_Z^\perp\), \begin{align} |\gamma + C_Z^\perp \rangle = \frac{1}{\sqrt{|C_Z^\perp|}} \sum_{\eta \in C_Z^\perp} |\gamma + \eta\rangle. \tag*{(3)}\end{align}