Galois-qudit CSS code[1][2]


An \([[n,k,d]]_{GF(q)} \) Galois-qudit true stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type Galois-qudit Pauli strings. The stabilizer generator matrix, taking values from \(GF(q)\), is of the form \begin{align} H=\begin{pmatrix}0 & H_{Z}\\ H_{X} & 0 \end{pmatrix} \label{eq:parityg} \end{align} such that the rows of the two blocks must be orthogonal \begin{align} H_X H_Z^T=0~. \label{eq:commG} \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 \(H_Z\).

Encoding is based on two related \(q\)-ary linear codes, an \([n,k_X,d_X]_q \) code \(C_X\) and \([n,k_Z,d_Z]_q \) 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:parityg} is the parity-check matrix of the code \(C_X\) (\(C_Z\)). The requirement \(C_X^\perp \subseteq C_Z\) guarantees \eqref{eq:commG}. 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}


Detects errors on \(d-1\) qubits, corrects errors on \(\left\lfloor (d-1)/2 \right\rfloor\) qubits.




Zoo code information

Internal code ID: galois_css

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Zoo Code ID: galois_css

Cite as:
“Galois-qudit CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_galois_css, title={Galois-qudit CSS code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004). DOI; quant-ph/0312164
J.-L. Kim and J. Walker, “Nonbinary quantum error-correcting codes from algebraic curves”, Discrete Mathematics 308, 3115 (2008). DOI
A. Niehage, “Nonbinary Quantum Goppa Codes Exceeding the Quantum Gilbert-Varshamov Bound”, Quantum Information Processing 6, 143 (2006). DOI

Cite as:

“Galois-qudit CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.