[Jump to code hierarchy]

Hyperbolic surface code

Description

An extension of the Kitaev surface code construction to hyperbolic manifolds. Given a cellulation of a manifold, qubits are put on \(i\)-dimensional faces, \(X\)-type stabilizers are associated with \((i-1)\)-faces, while \(Z\)-type stabilizers are associated with \(i+1\)-faces.

Protection

Constructions (see code children below) have yielded distances scaling favorably with the number of qubits. The use of hyperbolic surfaces allows one to circumvent bounds on code parameters (such as the BPT bound) that are valid for lattice geometries.

Gates

\((1,D-1)\) surface codes on hyperbolic geometries admit a fault-tolerant implementation of \(C^D Z\) gates [1].Higher-dimensional hyperbolic surface codes can admit a cup product structure and can thus have logical gates in the Clifford hierarchy implemented by constant-depth Clifford circuits [2].

Decoding

Hastings decoder [3].

Cousins

  • Holographic tensor-network code— Both holographic tensor-network and hyperbolic surface codes utilize tesselations of hyperbolic surfaces. Encodings for the former are hyperbolically tiled tensor networks, while the latter is defined on hyperbolically tiled physical-qubit lattices.
  • Single-shot code— A 4D hyperbolic surface code can be decoded with the Hastings decoder [3] in time \(O(n\log n)\) and with a logical error scaling inverse polynomially with \(n\).
  • Hyperbolic color code

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
Parents
Homological code
Hyperbolic surface code
Children
2D hyperbolic surface code
Guth-Lubotzky code
Freedman-Meyer-Luo code

References

[1]
Y. Wang, Y. Wang, Y.-A. Chen, W. Zhang, T. Zhang, J. Hu, W. Chen, Y. Gu, and Z.-W. Liu, “Efficient fault-tolerant implementations of non-Clifford gates with reconfigurable atom arrays”, npj Quantum Information 10, (2024) arXiv:2312.09111 DOI
[2]
N. P. Breuckmann, M. Davydova, J. N. Eberhardt, and N. Tantivasadakarn, “Cups and Gates I: Cohomology invariants and logical quantum operations”, (2024) arXiv:2410.16250
[3]
M. B. Hastings, “Decoding in Hyperbolic Spaces: LDPC Codes With Linear Rate and Efficient Error Correction”, (2013) arXiv:1312.2546
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: hyperbolic_surface

Cite as:
“Hyperbolic surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hyperbolic_surface
BibTeX:
@incollection{eczoo_hyperbolic_surface, title={Hyperbolic surface code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hyperbolic_surface} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/hyperbolic_surface

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/surface/hyperbolic/hyperbolic_surface.yml.