Hypersphere product code[1] 

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  • Quantum locally testable code (QLTC) — The hypersphere product code family has asymptotically diminishing soundness that scales as order \(O(1/\log (n)^2)\), locality of stabilizer generators scaling as order \(O(\log n/ \log\log n)\), and distance \(\Theta(\sqrt{n})\). Application of the generalized distance balancing scheme [2] using an asymptotically good classical code of length \(t\) yields \(O(1/(\log(n)^2 t^2))\) soundness and \(\Theta(\sqrt{n}t)\) distance while maintaining locality scaling and at the expense of a dimension scaling as \(\Theta(t^2)\) [3].

References

[1]
M. B. Hastings, “Quantum Codes from High-Dimensional Manifolds”, (2016) arXiv:1608.05089
[2]
S. Evra, T. Kaufman, and G. Zémor, “Decodable quantum LDPC codes beyond the \(\sqrt{n}\) distance barrier using high dimensional expanders”, (2020) arXiv:2004.07935
[3]
A. Wills, T.-C. Lin, and M.-H. Hsieh, “General Distance Balancing for Quantum Locally Testable Codes”, (2023) arXiv:2305.00689
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Zoo Code ID: hypersphere_product

Cite as:
“Hypersphere product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hypersphere_product
BibTeX:
@incollection{eczoo_hypersphere_product,
  title={Hypersphere product code},
  booktitle={The Error Correction Zoo},
  year={2022},
  editor={Albert, Victor V. and Faist, Philippe},
  url={https://errorcorrectionzoo.org/c/hypersphere_product}
}
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Permanent link:
https://errorcorrectionzoo.org/c/hypersphere_product

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/stabilizer/topological/surface/higher_dim_surface/hypersphere_product.yml.