Modular-qudit CSS code[13] 


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. 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 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 [4; 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 \(d\geq\min\{d_X,d_Z\}\). 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 \(\gamma \in C_X\), \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}


  • Modular-qudit stabilizer code — Modular-qudit CSS codes are modular-qudit stabilizer codes whose stabilizer groups admit a generating set of pure-\(X\) and pure-\(Z\) Pauli strings. Any \([[n,k,d]]_{\mathbb{Z}_q}\) stabilizer code can be mapped onto a \([[2n,2k,\geq d]]_{\mathbb{Z}_q}\) CSS code via symplectic doubling, which preserves geometric locality of a code up to a constant factor.
  • Group GKP code — An \(n\) modular-qubit CSS code corresponds to the \(\mathbb{Z}_q^{k_1} \subseteq \mathbb{Z}_q^{k_2} \subset \mathbb{Z}_q^{n}\) group construction, where \(k=k_2/k_1\), and where the group operation is addition. This construction should be extendable to additive \(q\)-ary codes over \(\mathbb{Z}_q\) since those are also groups under addition.
  • Calderbank-Shor-Steane (CSS) stabilizer code




A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist”, Physical Review A 54, 1098 (1996) arXiv:quant-ph/9512032 DOI
A. M. Steane, “Error Correcting Codes in Quantum Theory”, Physical Review Letters 77, 793 (1996) DOI
“Multiple-particle interference and quantum error correction”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 452, 2551 (1996) arXiv:quant-ph/9601029 DOI
S. Novák, “Homological Quantum Error Correction with Torsion”, (2024) arXiv:2405.03559
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Zoo Code ID: qudit_css

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“Modular-qudit CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_qudit_css, title={Modular-qudit CSS code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Modular-qudit CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.