# 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].

## Parent

- Generalized homological product CSS code — Balanced product codes result from a tensor product of two classical-code chain complexes, followed by a factoring out of certain symmetries.

## Children

- Dinur-Hsieh-Lin-Vidick (DHLV) code
- Fiber-bundle code — Fiber-bundle codes can be formulated in terms of a balanced product [1].
- Lifted-product (LP) code

## Cousins

- Subsystem qubit stabilizer code — Distance balancing is used to form balanced-product subsystem codes [1].
- Distance-balanced code — Distance balancing is used to form balanced-product subsystem codes [1].
- Left-right Cayley complex code — Left-right Cayley complexes can be obtained via a balanced product of \(G\)-graphs [4].

## References

- [1]
- N. P. Breuckmann and J. N. Eberhardt, “Balanced Product Quantum Codes”, IEEE Transactions on Information Theory 67, 6653 (2021) arXiv:2012.09271 DOI
- [2]
- P. Panteleev and G. Kalachev, “Asymptotically Good Quantum and Locally Testable Classical LDPC Codes”, (2022) arXiv:2111.03654
- [3]
- P. Panteleev and G. Kalachev, “Degenerate Quantum LDPC Codes With Good Finite Length Performance”, Quantum 5, 585 (2021) arXiv:1904.02703 DOI
- [4]
- I. Dinur et al., “Locally Testable Codes with constant rate, distance, and locality”, (2021) arXiv:2111.04808

## Page edit log

- Victor V. Albert (2022-01-03) — most recent
- Finnegan Voichick (2021-12-18)
- Nikolas Breuckmann (2021-12-14)

## Cite as:

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