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.
Gates
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].
Parent
Child
- \([[9,1,3]]\) Shor code — The Shor code is one of the nine-qubit surface codes defined on the projective plane \(\mathbb{R}P^2\) [1; Fig. 4].
Cousins
- 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].
References
- [1]
- M. H. Freedman and D. A. Meyer, “Projective plane and planar quantum codes”, (1998) arXiv:quant-ph/9810055
- [2]
- R. Kobayashi and G. Zhu, “Cross-Cap Defects and Fault-Tolerant Logical Gates in the Surface Code and the Honeycomb Floquet Code”, PRX Quantum 5, (2024) arXiv:2310.06917 DOI
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