# Balanced product (BP) 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.

## 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 whose size is of order \(\Theta(n)\) has a constant encoding rate [1].

## Decoding

BP-OSD decoder [2].

## Parents

- Galois-qudit CSS code
- 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

- Fiber-bundle code — Fiber-bundle codes can be formulated in terms of a balanced product [1].
- Lossless expander balanced-product code
- Lifted-product (LP) code — Coarsely speaking, a lifted product is a balanced product where the group \(G\) acts freely. However, in principle, a lifted product can be defined for rings that are more general than group algebras \( \mathbb{F}_q G \).

## 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 [3].
- X-cube model code — Generalized X-cube models [4] are constructed from a balanced product of the quantum repetion (1D Ising) code and the Newman-Moore code.
- Dinur-Hsieh-Lin-Vidick (DHLV) code — DHLV codes can be obtained from a balanced product of two expander codes [5].
- Toric code — Twisted toric codes can be obtained from balanced products of cyclic graphs over a cyclic group [1; Fig. 8].

## 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, “Degenerate Quantum LDPC Codes With Good Finite Length Performance”, Quantum 5, 585 (2021) arXiv:1904.02703 DOI
- [3]
- I. Dinur et al., “Locally Testable Codes with constant rate, distance, and locality”, (2021) arXiv:2111.04808
- [4]
- T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes II. Product constructions”, (2024) arXiv:2402.16831
- [5]
- P. Panteleev and G. Kalachev, “Maximally Extendable Sheaf Codes”, (2024) arXiv:2403.03651

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