Quantum Gabidulin code[1]
Description
A Galois-qudit stabilizer code over \(n\) Galois qudits of dimension \(q = 2^n \) that is useful in protecting against faults in qubit Clifford circuits acting on stacked quantum memories. This code can be treated as a code on an \(n\times n\) qubit stacked memory by decomposing each Galois qudit into a Kronecker product of \(n\) qubits; see [2,4][3; Sec. 5.3].
A quantum Gabidulin code is defined using two Gabidulin codes with associated parameters \(r,s\), respectively, such that \(r+s = n\) [1].
Protection
The code distance is the minimum rank distance --- the rank of the field element of the lowest-rank undetectable Galois-qudit Pauli error, with the rank calculated by writing the element as an \(n\times n\) binary matrix. The code is useful in protecting against faults in \(n\)-qubit Clifford circuits with \(n\) layers, which preserve the minimum rank distance.
Parent
Cousins
- Gabidulin code — A quantum Gabidulin code is defined using two Gabidulin codes with associated parameters \(r,s\), respectively, such that \(r+s = n\) [1].
- Rank-metric code — Quantum Gabidulin code and (classical) rank-metric code distances are based on ranks of the matrix representations of their corresponding errors.
References
- [1]
- N. Delfosse and G. Zémor, “Correction of circuit faults in a stacked quantum memory using rank-metric codes”, (2024) arXiv:2411.09173
- [2]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [3]
- A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
- [4]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
Page edit log
- Victor V. Albert (2024-12-10) — most recent
Cite as:
“Quantum Gabidulin code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quantum_gabidulin