## Description

Hyperbolic surface codes based on a tessellation of a closed 2D manifold with a hyperbolic geometry (i.e., non-Euclidean geometry, e.g., saddle surfaces when defined on a 2D plane).

For a tessellation involving regular polygons with \( r \) sides and \( s \) polygons meeting at each edge, the number of logical qubits is given by \( k = (1-2/r - 2/s) n + 2 \). Some possible tilings are \( \{r,s\}: \{7,3\}, \{5,4\} \). The weight of the stabilizer generators are dependent on the tiling, with \(\{5,4\}\) having lower weight than \(\{7,3\}\).

A semi-hyperbolic surface code [4] is a code defined on a \(\{4,s\}\) tiling, but where each square is replaced with a square region of a 2D lattice.

## Protection

## Rate

## Decoding

## Code Capacity Threshold

## Threshold

## Notes

## Parents

## Child

## Cousins

- Asymmetric quantum code — Asymmetric 2D hyperbolic surface codes have been constructed [11].
- Subsystem hyperbolic surface code

## References

- [1]
- C. D. Albuquerque, R. Palazzo Jr., and E. B. Silva, “Topological quantum codes on compact surfaces with genus g≥2”, Journal of Mathematical Physics 50, (2009) DOI
- [2]
- C. D. Albuquerque, R. Palazzo Jr., and E. B. Silva, “New classes of TQC associated with self-dual, quasi self-dual and denser tessellations”, Quantum Information and Computation 10, 956 (2010) DOI
- [3]
- N. P. Breuckmann and B. M. Terhal, “Constructions and Noise Threshold of Hyperbolic Surface Codes”, IEEE Transactions on Information Theory 62, 3731 (2016) arXiv:1506.04029 DOI
- [4]
- N. P. Breuckmann et al., “Hyperbolic and semi-hyperbolic surface codes for quantum storage”, Quantum Science and Technology 2, 035007 (2017) arXiv:1703.00590 DOI
- [5]
- O. Higgott and N. P. Breuckmann, “Subsystem Codes with High Thresholds by Gauge Fixing and Reduced Qubit Overhead”, Physical Review X 11, (2021) arXiv:2010.09626 DOI
- [6]
- A. A. Kovalev et al., “Numerical and analytical bounds on threshold error rates for hypergraph-product codes”, Physical Review A 97, (2018) arXiv:1804.01950 DOI
- [7]
- Y. Jiang et al., “Duality and free energy analyticity bounds for few-body Ising models with extensive homology rank”, Journal of Mathematical Physics 60, (2019) arXiv:1805.00644 DOI
- [8]
- A. A. Kovalev and L. P. Pryadko, “Fault tolerance of quantum low-density parity check codes with sublinear distance scaling”, Physical Review A 87, (2013) arXiv:1208.2317 DOI
- [9]
- N. P. Breuckmann and J. N. Eberhardt, “Quantum Low-Density Parity-Check Codes”, PRX Quantum 2, (2021) arXiv:2103.06309 DOI
- [10]
- N. Delfosse and G. Zémor, “Upper Bounds on the Rate of Low Density Stabilizer Codes for the Quantum Erasure Channel”, (2012) arXiv:1205.7036
- [11]
- C. D. de Albuquerque et al., “Euclidean and Hyperbolic Asymmetric Topological Quantum Codes”, (2021) arXiv:2105.01144

## Page edit log

- Victor V. Albert (2021-12-29) — most recent
- Elizabeth R. Bennewitz (2021-12-12)

## Cite as:

“2D hyperbolic surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/two_dimensional_hyperbolic_surface