Projective-plane surface code[1] 


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.


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 \(\mathcal{C}^*\).


The rate is \(1/n\), where \(n\) is the number of edges of the particular cellulation.


Fault-tolerant Hadamard gate [2].Complete logical gate set for a stack of projective-plane surface codes [2].

Fault Tolerance

Fault-tolerant Hadamard gate [2].




  • Honeycomb Floquet code — Implementing the honeycomb Floquet code on a non-orientable cross-cap geometry allows for a logical-\(HZ\) gate to be implemented via a measurement schedule [2].
  • \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code — The qudit Shor code is a small qudit surface code on a Möbius strip with smooth boundary, which is obtained from removing a face of the tesselation of the projective plane \(\mathbb{R}P^2\) [1; Fig. 4].


M. H. Freedman and D. A. Meyer, “Projective plane and planar quantum codes”, (1998) arXiv:quant-ph/9810055
R. Kobayashi and G. Zhu, “Fault-tolerant logical gates via constant depth circuits and emergent symmetries on non-orientable topological stabilizer and Floquet codes”, (2023) arXiv:2310.06917
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Zoo Code ID: real_projective_plane

Cite as:
“Projective-plane surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021.
@incollection{eczoo_real_projective_plane, title={Projective-plane surface code}, booktitle={The Error Correction Zoo}, year={2021}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Projective-plane surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021.