# Two-dimensional hyperbolic surface code[1]

## 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 \({7,3}\).

## Protection

Protects against Pauli errors with distance \( d \propto \log(n) \). Code parameters are \( [[n, (1-2/r - 2/s) n + 2, \log(n) ]] \)

## Rate

Two-dimensional hyperbolic surface codes have an asymptotically constant encoding rate \( k/n \) with a distance scaling logarithmically with \( n\) when the surface is closed. The encoding rate depends on the tiling \( {r,s} \) and is given by \( k/n = (1-2/r - 2/s) + 2/n \), which approaches a constant value as the number of physical qubits grows. The weight of the stabilizers is \( r \) for \( Z \)-checks and \( s \) for \( X \)-checks. For open boundary conditions, the code reduces to constant distnace.

## Decoding

Due to the symmetries of hyperbolic surface codes, optimal measurement schedules of the stabilizers can be found [2].

## Threshold

1\(\%\) - 5\(\%\) for a \({5,4}\) tiling under minimum-weight decoding [3]. For larger tilings, the lower bound on the distance decreases, suggesting the threshold will also decrease.

## Notes

See Sec III A of Ref. [4] for a description of this code.Connection to percolation theory as shown in [5].

## Parent

## Zoo code information

## References

- [1]
- N. P. Breuckmann and B. M. Terhal, “Constructions and Noise Threshold of Hyperbolic Surface Codes”, IEEE Transactions on Information Theory 62, 3731 (2016). DOI; 1506.04029
- [2]
- O. Higgott and N. P. Breuckmann, “Subsystem Codes with High Thresholds by Gauge Fixing and Reduced Qubit Overhead”, Physical Review X 11, (2021). DOI; 2010.09626
- [3]
- 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). DOI; 1208.2317
- [4]
- N. P. Breuckmann and J. N. Eberhardt, “Quantum Low-Density Parity-Check Codes”, PRX Quantum 2, (2021). DOI; 2103.06309
- [5]
- Nicolas Delfosse and Gilles Zémor, “Upper Bounds on the Rate of Low Density Stabilizer Codes for the Quantum Erasure Channel”. 1205.7036

## Cite as:

“Two-dimensional hyperbolic surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/two_dimensional_hyperbolic_surface