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].Gates
Logical gates via Dehn twists for balanced products of cyclic codes [2].Decoding
BP-OSD decoder [3].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].
- X-cube model code— Generalized X-cube models [5] 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 [6].
- Bivariate bicycle (BB) code— The \([[90,8,10]]\) bivariate bicycle code can be formulated as a balanced product of two cyclic codes [2].
- 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 expander codes with RS inner codes yield \([q^{\text{polylog}(q)},k\geq n^{1-\epsilon},n/\text{poly}(\log n)]_q\) LTCs exhibiting the multiplication property [7].
Primary Hierarchy
Parents
Generalized homological-product CSS codeGeneralized homological-product QLDPC 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 \( \mathbb{F}_q G \).
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]
- T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes II. Product constructions”, (2024) arXiv:2402.16831
- [6]
- P. Panteleev and G. Kalachev, “Maximally Extendable Sheaf Codes”, (2024) arXiv:2403.03651
- [7]
- L. Golowich and T.-C. Lin, “Quantum LDPC Codes with Transversal Non-Clifford Gates via Products of Algebraic Codes”, (2024) arXiv:2410.14662
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