Modular-qudit CSS code[1][2]

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}

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Zoo code information

Internal code ID: qudit_css

Your contribution is welcome!

on github.com (edit & pull request)

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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} }
Permanent link:
https://errorcorrectionzoo.org/c/qudit_css

References

[1]
Alexei Ashikhmin and Emanuel Knill, “Nonbinary Quantum Stabilizer Codes”. quant-ph/0005008
[2]
Avanti Ketkar et al., “Nonbinary stabilizer codes over finite fields”. quant-ph/0508070

Cite as:

“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.