## 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].

## Decoding

Hastings decoder [2].

## Parent

## Children

## 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 [2] in time \(O(n\log n)\) and with a logical error scaling inverse polynomially with \(n\).
- Hyperbolic color code

## References

- [1]
- Y.-F. Wang et al., “Efficient fault-tolerant implementations of non-Clifford gates with reconfigurable atom arrays”, (2024) arXiv:2312.09111
- [2]
- M. B. Hastings, “Decoding in Hyperbolic Spaces: LDPC Codes With Linear Rate and Efficient Error Correction”, (2013) arXiv:1312.2546

## Page edit log

- Victor V. Albert (2022-01-07) — most recent

## Cite as:

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