2D hyperbolic surface code

2D hyperbolic surface code[1–3]

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

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

2D 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 [5].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 [6].Two flag-based decoders [7].

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 [8].\(1.3\%\) for a phenomenological noise model for the \(\{4,5\}\)-hyperbolic surface code [4].

Threshold

1\(\%\) - 5\(\%\) for a \({5,4}\) tiling under minimum-weight decoding [9]. For larger tilings, the lower bound on the distance decreases, suggesting the threshold will also decrease.

Notes

See Sec III A of Ref. [10] for a description of this code.Connection to percolation theory as shown in [11].

Cousins

Asymmetric quantum code— Asymmetric 2D hyperbolic surface codes have been constructed [12].
Subsystem hyperbolic surface code

Member of code lists

Primary Hierarchy

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, C. Vuillot, E. Campbell, A. Krishna, and B. M. Terhal, “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, S. Prabhakar, I. Dumer, and L. P. Pryadko, “Numerical and analytical bounds on threshold error rates for hypergraph-product codes”, Physical Review A 97, (2018) arXiv:1804.01950 DOI
[7]: S. Vittal, A. Javadi-Abhari, A. W. Cross, L. S. Bishop, and M. Qureshi, “Flag Proxy Networks: Tackling the Architectural, Scheduling, and Decoding Obstacles of Quantum LDPC codes”, (2024) arXiv:2409.14283
[8]: Y. Jiang, I. Dumer, A. A. Kovalev, and L. P. Pryadko, “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
[9]: 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
[10]: N. P. Breuckmann and J. N. Eberhardt, “Quantum Low-Density Parity-Check Codes”, PRX Quantum 2, (2021) arXiv:2103.06309 DOI
[11]: N. Delfosse and G. Zémor, “Upper Bounds on the Rate of Low Density Stabilizer Codes for the Quantum Erasure Channel”, (2012) arXiv:1205.7036
[12]: C. D. de Albuquerque, G. G. La Guardia, R. Palazzo Jr., C. R. de O. Q. Queiroz, and V. L. Vieira, “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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/surface/hyperbolic/2d_hyperbolic/two_dimensional_hyperbolic_surface.yml.