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

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

Protection

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

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 [3].Bounds on code capacity thresholds using ML decoding can be obtained by mapping the effect of noise on the code to a statistical mechanical model [4].

Code Capacity Threshold

Bounds on code capacity thresholds using ML decoding can be obtained by mapping the effect of noise on the code to a statistical mechanical model [5].\(1.3\%\) for a phenomenological noise model for the \(\{4,5\}\)-hyperbolic surface code [2].

Threshold

1\(\%\) - 5\(\%\) for a \({5,4}\) tiling under minimum-weight decoding [6]. For larger tilings, the lower bound on the distance decreases, suggesting the threshold will also decrease.

Notes

See Sec III A of Ref. [7] for a description of this code.Connection to percolation theory as shown in [8].

Parent

Child

Cousin

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) arXiv:1506.04029 DOI
[2]
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
[3]
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
[4]
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
[5]
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
[6]
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
[7]
N. P. Breuckmann and J. N. Eberhardt, “Quantum Low-Density Parity-Check Codes”, PRX Quantum 2, (2021) arXiv:2103.06309 DOI
[8]
N. Delfosse and G. Zémor, “Upper Bounds on the Rate of Low Density Stabilizer Codes for the Quantum Erasure Channel”, (2012) arXiv:1205.7036
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Zoo Code ID: two_dimensional_hyperbolic_surface

Cite as:
“Two-dimensional hyperbolic surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/two_dimensional_hyperbolic_surface
BibTeX:
@incollection{eczoo_two_dimensional_hyperbolic_surface, title={Two-dimensional hyperbolic surface code}, booktitle={The Error Correction Zoo}, year={2021}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/two_dimensional_hyperbolic_surface} }
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“Two-dimensional hyperbolic surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/two_dimensional_hyperbolic_surface

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