# Hypersphere product code[1]

## Description

Homological code based on products of hyperspheres. 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 \(\Theta(\sqrt{n})\).

## Parent

## Cousins

- 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 generalized distance balancing [2] to hemicubic codes 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].
- Distance-balanced code — 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 generalized distance balancing [2] to hemicubic codes 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

## Page edit log

- Victor V. Albert (2022-09-26) — most recent

## Cite as:

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