Description
An \([[n,k,d]]_q \) Galois-qudit true stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type Galois-qudit Pauli strings. Codes can be defined from chain complexes over \(GF(q)\) via an extension of qubit CSS-to-homology correspondence to Galois qudits.
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} \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:commG} \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 \(H_Z\).
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 \(H\) (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}
Protection
Parents
- True Galois-qudit stabilizer code — Galois-qudit CSS codes are true stabilizer codes [7].
- Calderbank-Shor-Steane (CSS) stabilizer code
Children
- Qubit CSS code — Galois-qudit CSS codes for \(q=2\) are (qubit) CSS codes.
- Binary quantum Goppa code — Goppa codes can be realized in the CSS code construction [8].
- Folded quantum Reed-Solomon (FQRS) code — Folding an quantum polynomial code on \(q\)-dimensional Galois qudits yields an FQRS code on \(q^m\)-dimensional Galois qudits.
- Balanced product (BP) code
- Distance-balanced code
- Galois-qudit topological code
- Generalized bicycle (GB) code — A GB code is a Galois-qudit CSS code constructed from a pair of equivalent index-two quasi-cyclic linear codes.
- Approximate secret-sharing code — The code required to construct this code must be a non-degenerate Galois-qubit CSS code.
- Singleton-bound approaching AQECC
- Skew-cyclic CSS code
Cousins
- Linear \(q\)-ary code — Construction uses two related \(q\)-ary linear codes \(C_X\) and \(C_Z\).
- Cyclic linear \(q\)-ary code — Galois CSS codes can be constructed using self-orthogonal \(q\)-ary cyclic codes [9].
- Galois-qudit BCH code — Some Galois-qudit BCH codes are CSS.
- Galois-qudit GRS code — Some Galois-qudit GRS codes are CSS.
References
- [1]
- A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist”, Physical Review A 54, 1098 (1996) arXiv:quant-ph/9512032 DOI
- [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) arXiv:quant-ph/9601029 DOI
- [4]
- E. Knill, “Group Representations, Error Bases and Quantum Codes”, (1996) arXiv:quant-ph/9608049
- [5]
- M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI
- [6]
- J.-L. Kim and J. Walker, “Nonbinary quantum error-correcting codes from algebraic curves”, Discrete Mathematics 308, 3115 (2008) DOI
- [7]
- D. Gottesman. Surviving as a quantum computer in a classical world
- [8]
- A. Niehage, “Nonbinary Quantum Goppa Codes Exceeding the Quantum Gilbert-Varshamov Bound”, Quantum Information Processing 6, 143 (2006) DOI
- [9]
- Y. Tang et al., “New quantum codes from dual-containing cyclic codes over finite rings”, (2016) arXiv:1608.06674
Page edit log
- Victor V. Albert (2022-09-28) — most recent
- Leonid Pryadko (2022-02-16)
- Daniel Gottesman (2022-02-16)
- Victor V. Albert (2022-02-16)
- Qingfeng (Kee) Wang (2022-01-07)
Cite as:
“Galois-qudit CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_css