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 [8; Def. 2], 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 Galois qudits 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). Specializing to the case when \(C_Z=[n,k,d]_q\) is dual-containing yields a \([[n,2k-n,\geq d_Z]]_q\) Galois-qudit CSS code with \(C_X = C_Z^\perp\). 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}
Galois-qudit CSS codes can also be understood in terms graphs via the reflexive stabilizer framework, which also allows one to define a code for a given set of Pauli errors [9].
Protection
Detects errors on \(d-1\) qubits, corrects errors on \(\left\lfloor (d-1)/2 \right\rfloor\) qubits.
An \([[n,k,d]]_q\) CSS code can be propagated to an \([[n-2,k,d-1]]_q\) code [10].
Parents
- True Galois-qudit stabilizer code — Galois-qudit CSS codes are true stabilizer codes [11].
- Group GKP code — An \(n\) Galois-qubit CSS code corresponds to the \(GF(q)^{k_1} \subseteq GF(q)^{k_2} \subset GF(q)^{n}\) 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 since those are also groups under addition.
- Calderbank-Shor-Steane (CSS) stabilizer code
Children
- Qubit CSS code — Galois-qudit CSS codes for \(q=2\) are (qubit) CSS 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
- Two-block CSS code
- Quantum quadratic-residue (QR) code
- Quantum Goppa code — Goppa codes can be realized in the CSS code construction [12].
- Quantum Tamo-Barg (QTB) code
- 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
Cousins
- Linear \(q\)-ary code — The Galois-qudit CSS 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 [13].
- Quantum locally recoverable code (QLRC) — A Galois-qudit CSS code is a QLRC of locality \(r\) if each qudit participates in at least one \(X\)-type and one \(Z\)-type stabilizer whose product is of weight \(\leq r\) [14; Corr. 34].
- Galois-qudit BCH code — Galois-qudit BCH codes can be constructed via the CSS construction or the Hermitian construction.
- Quantum duadic code — Quantum duadic codes can be constructed via the CSS construction or the Hermitian construction.
- Galois-qudit quantum RM code — Galois-qudit RM codes can be constructed via the CSS construction or the Hermitian construction.
- Galois-qudit GRS code — Galois-qudit GRS codes can be constructed via the CSS construction or the Hermitian construction.
- Subsystem Galois-qudit CSS code — Subsystem Galois-qudit CSS codes reduce to (subspace) Galois-qudit CSS codes when there is no gauge subsystem.
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]
- R. Matsumoto and T. Uyematsu, “Constructing quantum error-correcting codes for p^m-state systems from classical error-correcting codes”, (2000) arXiv:quant-ph/9911011
- [6]
- M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI
- [7]
- J.-L. Kim and J. Walker, “Nonbinary quantum error-correcting codes from algebraic curves”, Discrete Mathematics 308, 3115 (2008) DOI
- [8]
- M. GRASSL, M. RÖTTELER, and T. BETH, “EFFICIENT QUANTUM CIRCUITS FOR NON-QUBIT QUANTUM ERROR-CORRECTING CODES”, International Journal of Foundations of Computer Science 14, 757 (2003) arXiv:quant-ph/0211014 DOI
- [9]
- R. R. Vandermolen and D. Wright, “Graph-theoretic approach to quantum error correction”, Physical Review A 105, (2022) arXiv:2110.08414 DOI
- [10]
- M. Grassl, “New quantum codes from CSS codes”, Quantum Information Processing 22, (2023) arXiv:2208.05353 DOI
- [11]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [12]
- A. Niehage, “Nonbinary Quantum Goppa Codes Exceeding the Quantum Gilbert-Varshamov Bound”, Quantum Information Processing 6, 143 (2006) DOI
- [13]
- Y. Tang et al., “New quantum codes from dual-containing cyclic codes over finite rings”, (2016) arXiv:1608.06674
- [14]
- L. Golowich and V. Guruswami, “Quantum Locally Recoverable Codes”, (2023) arXiv:2311.08653
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