Balanced product (BP) code[1]
Description
Family of CSS quantum codes obtained from two classical-code chain complexes that share a common group symmetry. The balanced product can be understood as taking the usual tensor or hypergraph product and then quotienting by the shared symmetry action. This can reduce the overall number of physical qubits \(n\) while, in favorable cases, preserving the number of encoded qubits and the code distance, thereby improving the encoding rate \(k/n\) and normalized distance \(d/n\) compared to the underlying tensor or hypergraph product.
For trivial group action, the construction reduces to a hypergraph product code. For cyclic groups, it overlaps with fiber-bundle and lifted-product constructions [1].
Rate
The original explicit balanced-product family is first constructed as a horizontal subsystem code with \(k \in \Theta(n^{2/3})\), \(d_X \in \Omega(n^{1/3})\), and \(d_Z \in \Theta(n)\); after distance balancing, it yields an LDPC family with \(k \in \Theta(n^{4/5})\) and \(d \in \Omega(n^{3/5})\) [1]. For balanced products of two good classical LDPC codes over groups of order \(\Theta(n)\), the original paper proves constant encoding rate and conjectures linear distance [1].Gates
Logical gates via Dehn twists for balanced products of cyclic codes [2].Decoding
BP-OSD decoder [3].Cousins
- Subsystem qubit stabilizer code— The original explicit balanced-product family is first constructed as a horizontal subsystem balanced-product code built from expander codes and cyclic repetition codes [1].
- Distance-balanced code— Applying distance balancing to the explicit subsystem balanced-product family of Ref. [1] yields an LDPC code family with \(k \in \Theta(n^{4/5})\) and \(d \in \Omega(n^{3/5})\).
- Left-right Cayley complex code— Left-right Cayley complexes can be obtained via a balanced product of \(G\)-graphs [4].
- Fibonacci fractal spin-liquid code— The Fibonacci fractal spin-liquid code is a hypergraph product of the repetition code and the Fibonacci code [5], and can be formulated directly as a BP code [6].
- \([[90,8,10]]\) BB6 code— The \([[90,8,10]]\) BB code can be formulated as a balanced product of two cyclic codes [2].
- Dinur-Hsieh-Lin-Vidick (DHLV) code— DHLV codes can be obtained from a balanced product of two expander codes [7].
- Toric code— Twisted toric codes can be obtained from balanced products of cyclic graphs over a cyclic group [1; Fig. 8].
- Galois-qudit expander code— Balanced products of the RS-based expander-code complexes in [8] yield \([n,k\geq n^{1-\epsilon},d\geq n/\operatorname{poly}(\log n)]_q\) LTCs exhibiting the multiplication property.
Primary Hierarchy
Parents
Generalized homological-product CSS codeGeneralized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Balanced product codes result from a tensor product of two classical-code chain complexes, followed by a factoring out of certain symmetries.
Balanced product (BP) code
Children
Fiber-bundle codes can be formulated in terms of a balanced product [1].
Coarsely speaking, a lifted product is a balanced product where the group \(G\) acts freely. In principle, a lifted product can be defined for rings that are more general than group algebras \( \mathbb{F}_q G \).
2BGA codes can be formulated as balanced product codes [9; Rem. C.1].
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]
- R. Tiew and N. P. Breuckmann, “Low-Overhead Entangling Gates from Generalised Dehn Twists”, (2024) arXiv:2411.03302
- [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, S. Evra, R. Livne, A. Lubotzky, and S. Mozes, “Locally Testable Codes with constant rate, distance, and locality”, (2021) arXiv:2111.04808
- [5]
- B. Yoshida, “Exotic topological order in fractal spin liquids”, Physical Review B 88, (2013) arXiv:1302.6248 DOI
- [6]
- Y. Tan, B. Roberts, N. Tantivasadakarn, B. Yoshida, and N. Y. Yao, “Fracton models from product codes”, Physical Review Research 7, (2025) arXiv:2312.08462 DOI
- [7]
- P. Panteleev and G. Kalachev, “Maximally Extendable Sheaf Codes”, (2024) arXiv:2403.03651
- [8]
- L. Golowich and T.-C. Lin, “Quantum LDPC Codes with Transversal Non-Clifford Gates via Products of Algebraic Codes”, (2024) arXiv:2410.14662
- [9]
- J. N. Eberhardt and V. Steffan, “Logical Operators and Fold-Transversal Gates of Bivariate Bicycle Codes”, (2024) arXiv:2407.03973
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