# Fiber-bundle code[1]

Also known as Twisted product code.

## Description

A CSS code constructed by combining one code as the base and another as the fiber of a fiber bundle. In particular, taking a random LDPC code as the base and a cyclic repetition code as the fiber yields, after distance balancing, a QLDPC code with distance \(\Omega(n^{3/5}\text{polylog}(n))\) and rate \(\Omega(n^{-2/5}\text{polylog}(n))\) is obtained.

## Rate

Rate \(k/n = \Omega(n^{-2/5}/\text{polylog}(n))\), distance \(d=\Omega(n^{3/5}/\text{polylog}(n))\). This is the first QLDPC code to achieve a distance scaling better than \(\sqrt{n}~\text{polylog}(n)\).

## Decoding

Greedy algorithm can be used to efficiently decode \(X\) errors, but no known efficient decoding of \(Z\) errors yet [1].

## Parents

- Generalized homological-product qubit CSS code
- Balanced product (BP) code — Fiber-bundle codes can be formulated in terms of a balanced product [2].

## Child

- Homological product code — Fiber-bundle code can be viewed as a homological product code with a twisted product.

## Cousins

- Lifted-product (LP) code — The specific fiber-bundle QLDPC code achieving a distance scaling better than \(\sqrt{n}~\text{polylog}(n)\) can also be formulated as an LP code (see published version [3]).
- Distance-balanced code — Fiber-bundle code constructions use distance balancing and weight reduction to increase distance.
- Random stabilizer code — Taking a random LDPC code as the base and a cyclic repetition code as the fiber yields, after distance balancing, a QLDPC code with distance \(\Omega(n^{3/5}\text{polylog}(n))\) and rate \(\Omega(n^{-2/5}\text{polylog}(n))\) is obtained.
- Lifted-product (LP) code — Lifted products of a length-one with a length-\(m\) chain complex can be thought of as fiber-bundle codes.

## References

- [1]
- M. B. Hastings, J. Haah, and R. O’Donnell, “Fiber bundle codes: breaking the n \({}^{\text{1/2}}\) polylog( n ) barrier for Quantum LDPC codes”, Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (2021) arXiv:2009.03921 DOI
- [2]
- N. P. Breuckmann and J. N. Eberhardt, “Balanced Product Quantum Codes”, IEEE Transactions on Information Theory 67, 6653 (2021) arXiv:2012.09271 DOI
- [3]
- M. B. Hastings, J. Haah, and R. O’Donnell, “Fiber bundle codes: breaking the n \({}^{\text{1/2}}\) polylog( n ) barrier for Quantum LDPC codes”, Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (2021) DOI

## Page edit log

- Victor V. Albert (2022-01-04) — most recent
- Jon Nelson (2021-12-15)

## Cite as:

“Fiber-bundle code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/fiber_bundle