Code limits

Kenta Kasai, Manabu Hagiwara, Hideki Imai +1 more

In this paper, we propose a construction method for two-level quantum error-correcting codes via non-binary LDPC codes over an extended field of order 2p, p an integer p >; 1. The proposed quantum error-correcting codes are based on binary quasi-cyclic LDPC codes which have almost achieved a “Bounded Distance Decoding (BDD)” limit but have not surpassed the limit yet. Quantum codes constructed from the proposed method surpass the BDD limit. Furthermore the codes outperform the efficiently-decodable state-of-the-art quantum codes.

Stefan Hollands, Ko Sanders

Simon Burton, Dan E. Browne

In a fault tolerant quantum computer, quantum codes are expected to serve the conflicting purposes of protecting quantum information while also allowing that information to be manipulated by fault-tolerant gates. We introduce a new technique for placing limitations on such gates, and apply this technique to a class of quantum codes known as hypergraph product codes contained within the vertical sector. These codes are constructed from input which is a pair of classical linear codes, and generalize the Kitaev surface code which is the hypergraph product of classical repetition codes. We provide a necessary condition on these input codes, under which the resulting hypergraph product code has transversal gates limited to the Clifford group. We conjecture that this condition is satisfied by all$[n,k,d]$Gallagher codes with$d\ge 3$and$k\le n/2$. This work is a generalization of an argument due to Bravyi and König, and we also conjecture this is a refinement of the recent notion of disjointness due to Jochym-O’Connoret al.

Ki-Moon Lee, Hayder Radha, Beom-Jin Kim +1 more

We present Kovalenko's full-rank limit as a tight probabilistic lower bound for error performances of LDPC codes and LT codes over BEC. From the limit, we derive a full-rank overhead as a tight lower-bound for stable overheads for successful maximum-likelihood decoding of the codes.

Richard Thompson