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

Child

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 quantum code
  • 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 \(N^{1/2} \operatorname{polylog}(N)\) Barrier for Quantum LDPC Codes”, (2020) arXiv:2009.03921
[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
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Zoo Code ID: fiber_bundle

Cite as:
“Fiber-bundle code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/fiber_bundle
BibTeX:
@incollection{eczoo_fiber_bundle, title={Fiber-bundle code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/fiber_bundle} }
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Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/qldpc/homological/balanced_product/fiber_bundle.yml.