Balanced product (BP) code[1] 


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.


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].


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].


BP-OSD decoder [3].





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

Cite as:
“Balanced product (BP) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
  title={Balanced product (BP) code},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
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Cite as:

“Balanced product (BP) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.