Decoding Across the Quantum LDPC Code Landscape

Abstract

We show that belief propagation combined with ordered statistics post-processing is a general decoder for quantum low density parity check codes constructed from the hypergraph product. To this end, we run numerical simulations of the decoder applied to three families of hypergraph product code: topological codes, fixed-rate random codes and a new class of codes that we call semi-topological codes. Our new code families share properties of both topological and random hypergraph product codes, with a construction that allows for a finely-controlled trade-off between code threshold and stabilizer locality. Our results indicate thresholds across all three families of hypergraph product code, and provide evidence of exponential suppression in the low error regime. For the Toric code, we observe a threshold in the range 9.9 ± 0.2%. This result improves upon previous quantum decoders based on belief propagation, and approaches the performance of the minimum weight perfect matching algorithm. We expect semi-topological codes to have the same threshold as Toric codes, as they are identical in the bulk, and we present numerical evidence supporting this observation.

Refined Belief Propagation Decoding of Sparse-Graph Quantum Codes

“…For nondegenerate quantum codes, their decoding behaviors are like classical codes and BP usually works well with the techniques of message normalization and message scheduling (see examples in [25,26]). However, BP fails to decode highly-degenerate quantum codes [33][34][35], such as the surface codes [8,9]. Next we will analyze the energy topology of a degenerate quantum code.…”

“…The neurons denoted by Γ n→m will compute messages λ Smn (Γ n→m ) but this is not explicitly shown. Note that there are additional edges (dotted curves) from ∆ m→n to λ Smn (Γ n→m ), which are not considered in the previous BP methods [11,25,[32][33][34], nor in the neural networks of the neural BP [38,49].…”

“…Conventional BP 4 (parallel BP 4 ) cannot decide whether an error X is on qubit 3 or qubit 4. To break this symmetry, possible methods include post-processes (random perturbation [32], OSD [33,34], etc) or pre-processes (like training [36][37][38][39]). On the other hand, using serial MBP 4 with α = 0.65 can make correct decision without additional processes.…”

“…(We say that a code has high degeneracy or is highly-degenerate if it has many low-weight stabilizers compared to its minimum distance.) A stabilizer code has its Tanner graph inevitably containing many short cycles [11,32], which deteriorate the message-passing process in BP, especially for codes with high degeneracy [33][34][35]. Any message-passing or neural network decoder may have this issue.…”

“…Any message-passing or neural network decoder may have this issue. Previously this issue is handled with additional efforts in pre-training by neural networks [36][37][38][39] or post-processing by the ordered statistics decoding [33,34]. In addition, their decoders treat Pauli X and Z errors separately, which may incur additional overhead or performance loss.…”