[Jump to code hierarchy]

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

Primary Hierarchy

Parents
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. However, 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

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/stabilizer/qldpc/balanced_product/balanced_product.yml.