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

Zoo code information

Internal code ID: real_projective_plane

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: real_projective_plane

Cite as:
“Projective-plane surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/real_projective_plane
BibTeX:
@incollection{eczoo_real_projective_plane, title={Projective-plane surface code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/real_projective_plane} }
Permanent link:
https://errorcorrectionzoo.org/c/real_projective_plane

References

[1]
Michael H. Freedman and David A. Meyer, “Projective plane and planar quantum codes”. quant-ph/9810055

Cite as:

“Projective-plane surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/real_projective_plane

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/surface/real_projective_plane.yml.