Projective-plane surface code[1]
Description
A family of Kitaev surface codes on the non-orientable 2-dimensional compact manifold \(\mathbb{R}P^2\) (in contrast to a genus-\(g\) surface). Whereas genus-\(g\) surface codes require \(2g\) logical qubits, qubit codes on \(\mathbb{R}P^2\) are made from a single logical qubit.
Protection
If \(\mathcal{C}\) is a cellulation of \(\mathbb{R}P^2\), then the bit-flip distance \(d_X\) is the shortest cycle in \(\mathcal{C}\), and the phase-flip distance \(d_Z\) is the shortest cycle in the dual cellulation \(\mathcal{C}^*\).
Rate
The rate is \(1/n\), where \(n\) is the number of edges of the particular cellulation.
Parent
Cousin
- \([[9,1,3]]\) Shor code — Shor's code is the smallest surface code defined on the projective plane [1].
References
- [1]
- M. H. Freedman and D. A. Meyer, “Projective plane and planar quantum codes”, (1998) arXiv:quant-ph/9810055
Page edit log
- Victor V. Albert (2021-12-16) — most recent
- Eric Kubischta (2021-12-15)
Cite as:
“Projective-plane surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/real_projective_plane