Alternative names: \(\mathbb{F}_q\)-qudit surface code.
Description
Extension of the surface code to 2D lattices of Galois qudits.Cousin
- Abelian quantum-double stabilizer code— A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [3,5–7][4; Sec. 5.3]. Galois-qudit surface codes yield Abelian quantum-double codes with \(GF(p^m)\cong \mathbb{Z}_p^m\) topological order via this decomposition.
Primary Hierarchy
Parents
Generalized homological-product CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [3,5–7][4; Sec. 5.3]. Galois-qudit surface codes yield Abelian quantum-double codes with \(GF(p^m)\cong \mathbb{Z}_p^m\) topological order via this decomposition.
Galois-qudit surface code
Children
The Galois-qudit surface code for \(q=2\) reduces to the surface code.
References
- [1]
- S. S. Bullock and G. K. Brennen, “Qudit surface codes and gauge theory with finite cyclic groups”, Journal of Physics A: Mathematical and Theoretical 40, 3481 (2007) arXiv:quant-ph/0609070 DOI
- [2]
- I. Andriyanova, D. Maurice, and J.-P. Tillich, “New constructions of CSS codes obtained by moving to higher alphabets”, (2012) arXiv:1202.3338
- [3]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [4]
- A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
- [5]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [6]
- Q. T. Nguyen, “Good binary quantum codes with transversal CCZ gate”, (2024) arXiv:2408.10140
- [7]
- L. Golowich and V. Guruswami, “Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates”, (2024) arXiv:2408.09254
Page edit log
- Victor V. Albert (2022-07-27) — most recent
Cite as:
“Galois-qudit surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_topological