Trapping Sets of Quantum LDPC Codes

Abstract

Iterative decoders for finite length quantum lowdensity parity-check (QLDPC) codes are attractive because their hardware complexity scales only linearly with the number of physical qubits. However, they are impacted by short cycles, detrimental graphical configurations known as trapping sets (TSs) present in a code graph as well as symmetric degeneracy of errors. These factors significantly degrade the decoder decoding probability performance, and cause so-called error floor. In this paper, we establish a systematic methodology by which one can identify and classify quantum trapping sets (QTSs) according to their topological structure and decoder used. Conventional definition of a TS from classical error correction is generalized to address the syndrome decoding scenario for QLDPC codes. We show that the knowledge of QTSs can be used to design better QLDPC code and decoder. Frame error rate improvements of two orders of magnitude in the error floor regime are demonstrated for some practical finite-length QLDPC codes without requiring any post-processing.

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.…”

“…Since a target δ m > 0 (a target ∆ m > 0), we may also consider another energy function to focus on negative ∆ m (i.e., to focus on mismatched checks) by where ẑm = Ê, S m . This energy function is used by a simplified BP called bit-flipping BP [30,31,35,78,79], which decides the update direction (which bit to flip) by minimizing the number of unmatched syndrome bits between z and ẑ. Bit-flipping BP has very low complexity and is useful for analyzing the convergence, since it only tracks the hard-decision information of the variable nodes. (Bit-flipping BP can also be used in practice, e.g., it was used in decoding expander codes [78].…”

“…(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.…”

Decoding of Quantum Data-Syndrome Codes via Belief Propagation

“…We first explain the serial schedule that will be conducted in the following simulations. In [24], it is demonstrated that based on the raw check matrix of the [[129, 28]] HP code, the parallel BP 4 decoding does not perform well due to decoding oscillation (which is caused by the numerous short cycles and symmetric sub-graphs in the Tanner graph [24], [31]); on the other hand, the serial BP 4 along variable nodes performs quite well by using the raw matrix [24]. We have created a check matrix so that each of its column has weight ≥ 2 for Theorem 2; however, the serial update along variable nodes is too aggressive at certain variable node for the new check matrix (when computing the hard-decision and outgoing messages).…”

“…3). The message update order plays an important role in BP, especially when the Tanner graph has numerous short cycles [24], [31].…”