Balanced product code[1]

Description

Family of CSS quantum codes based on products of two classical codes which share common symmetries. The balanced product can be understood as taking the usual tensor/hypergraph product and then factoring out the symmetries factored. This reduces the overall number of physical qubits \(n\), while, under certain circumstances, leaving the number of encoded qubits \(k\) and the code distance \(d\) invariant. This leads to a more favourable encoding rate \(k/n\) and normalized distance \(d/n\) compared to the tensor/hypergraph product.

Protection

Taking balanced products of two classical LDPC codes which have a symmetry group which grows linearly in their block lengths were known to give QLDPC codes with a linear rate and which were conjectured to have linear distance [1]. This conjecture was proved in Ref. [2].

Rate

A notable family of balanced product codes encode \(k \in \Theta(n^{4/5})\) logical qubits with distance \(d \in \Omega(n^{3/5})\) for any number of physical qubits \(n\). Additionally, it is known that the code constructed from the balanced product of two good classical LDPC codes over groups of order \(\Theta(n)\) has a constant encoding rate [1].

Decoding

BP-OSD decoder [3].

Parents

Children

Cousins

Zoo code information

Internal code ID: balanced_product

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: balanced_product

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

References

[1]
N. P. Breuckmann and J. N. Eberhardt, “Balanced Product Quantum Codes”, IEEE Transactions on Information Theory 67, 6653 (2021). DOI; 2012.09271
[2]
Pavel Panteleev and Gleb Kalachev, “Asymptotically Good Quantum and Locally Testable Classical LDPC Codes”. 2111.03654
[3]
P. Panteleev and G. Kalachev, “Degenerate Quantum LDPC Codes With Good Finite Length Performance”, Quantum 5, 585 (2021). DOI; 1904.02703

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/properties/qldpc/balanced_product.yml.