Modular-qudit CSS code[1][2][3]

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. 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} \end{align} such that the rows of the two blocks must be orthogonal \begin{align} H_X H_Z^T=0~. \label{eq:commQ} \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. 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 \(H\) \(\eqref{eq:parityq}\) is the parity-check matrix of the code \(C_X\) (\(C_Z\)). The requirement \(C_X^\perp \subseteq C_Z\) guarantees \(\eqref{eq:commQ}\). 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. \end{align}

Parent

Children

Cousins

  • Calderbank-Shor-Steane (CSS) stabilizer code — Extension of CSS codes to modular-integer qudits.
  • Linear \(q\)-ary code — Construction for prime \(q=p\) uses two related \(p\)-ary linear codes \(C_X\) and \(C_Z\).
  • 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 \(k=k_2/k_1\).

References

[1]
A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist”, Physical Review A 54, 1098 (1996). DOI; quant-ph/9512032
[2]
A. M. Steane, “Error Correcting Codes in Quantum Theory”, Physical Review Letters 77, 793 (1996). DOI
[3]
“Multiple-particle interference and quantum error correction”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 452, 2551 (1996). DOI; quant-ph/9601029
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Zoo code information

Internal code ID: qudit_css

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Zoo Code ID: qudit_css

Cite as:
“Modular-qudit CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_css
BibTeX:
@incollection{eczoo_qudit_css, title={Modular-qudit CSS code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qudit_css} }
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“Modular-qudit CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_css

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qudits/qudit_css.yml.