Fiber-bundle code

Fiber-bundle code[1]

Alternative names: 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 of order \(\Omega(n^{3/5}\text{polylog}(n))\) and rate of order \(\Omega(n^{-2/5}\text{polylog}(n))\) 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].

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 [2]). Lifted products of a length-one with a length-\(m\) chain complex can be thought of as fiber-bundle codes [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 of order \(\Omega(n^{3/5}\text{polylog}(n))\) and rate of order \(\Omega(n^{-2/5}\text{polylog}(n))\) is obtained.

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

Parents

Generalized homological-product qubit CSS code

Balanced product (BP) codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

Fiber-bundle codes can be formulated in terms of a balanced product [3].

Fiber-bundle code

Children

Homological product code

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

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 1276 (2021) arXiv:2009.03921 DOI
[2]: 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 1276 (2021) DOI
[3]: N. P. Breuckmann and J. N. Eberhardt, “Balanced Product Quantum Codes”, IEEE Transactions on Information Theory 67, 6653 (2021) arXiv:2012.09271 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

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