Projective-plane surface code

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 via constant-depth Clifford circuit [2].Complete logical gate set for a stack of projective-plane surface codes [2].

Fault Tolerance

Fault-tolerant Hadamard gate via constant-depth Clifford circuit [2].

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].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

Homological code

Kitaev surface codeCDSC Twist-defect surface Lattice stabilizer Generalized homological-product QLDPC CSS Stabilizer Qubit Abelian topological Topological Hamiltonian-based QECC Quantum

Parents

Kitaev surface code

The projective-plane surface code is the surface code on \(\mathbb{R}P^2\).

Projective-plane surface code

Children

\([[9,1,3]]\) 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][3; Fig. 20].

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
[3]: H. Bombin and M. A. Martin-Delgado, “Homological error correction: Classical and quantum codes”, Journal of Mathematical Physics 48, (2007) arXiv:quant-ph/0605094 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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/real_projective_plane.yml.