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Improved Decoding of Quantum Tanner Codes Using Generalized Check Nodes

Olai Å. Mostad, Eirik Rosnes, Hsuan-Yin Lin

TL;DR

It is shown that the proposed enhanced generalized BP decoder for quantum Tanner codes significantly outperforms the standard quaternary BP decoder with memory effects, as well as the recently proposed Relay-BP decoder, even outperforming generalized bicycle (GB) codes with comparable parameters in some cases.

Abstract

We study the decoding problem for quantum Tanner codes and propose to exploit the underlying local code structure by grouping check nodes into more powerful generalized check nodes for enhanced iterative belief propagation (BP) decoding by decoding the generalized checks using a maximum a posteriori (MAP) decoder as part of the check node processing of each decoding iteration. We mainly study the finite-length setting and show that the proposed enhanced generalized BP decoder for quantum Tanner codes significantly outperforms the standard quaternary BP decoder with memory effects, as well as the recently proposed Relay-BP decoder, even outperforming generalized bicycle (GB) codes with comparable parameters in some cases. For other classes of quantum low-density parity-check (qLDPC) codes, we propose a greedy algorithm to combine checks for generalized BP decoding. However, for GB codes, bivariate bicycle codes, hypergraph product codes, and lifted-product codes, there seems to be limited gain by combining simple checks into more powerful ones. To back up our findings, we also provide a theoretical cycle analysis for the considered qLDPC codes.

Improved Decoding of Quantum Tanner Codes Using Generalized Check Nodes

TL;DR

It is shown that the proposed enhanced generalized BP decoder for quantum Tanner codes significantly outperforms the standard quaternary BP decoder with memory effects, as well as the recently proposed Relay-BP decoder, even outperforming generalized bicycle (GB) codes with comparable parameters in some cases.

Abstract

We study the decoding problem for quantum Tanner codes and propose to exploit the underlying local code structure by grouping check nodes into more powerful generalized check nodes for enhanced iterative belief propagation (BP) decoding by decoding the generalized checks using a maximum a posteriori (MAP) decoder as part of the check node processing of each decoding iteration. We mainly study the finite-length setting and show that the proposed enhanced generalized BP decoder for quantum Tanner codes significantly outperforms the standard quaternary BP decoder with memory effects, as well as the recently proposed Relay-BP decoder, even outperforming generalized bicycle (GB) codes with comparable parameters in some cases. For other classes of quantum low-density parity-check (qLDPC) codes, we propose a greedy algorithm to combine checks for generalized BP decoding. However, for GB codes, bivariate bicycle codes, hypergraph product codes, and lifted-product codes, there seems to be limited gain by combining simple checks into more powerful ones. To back up our findings, we also provide a theoretical cycle analysis for the considered qLDPC codes.
Paper Structure (14 sections, 5 theorems, 16 equations, 4 figures, 1 table, 4 algorithms)

This paper contains 14 sections, 5 theorems, 16 equations, 4 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Graph Constructions
  5. Quantum Error-Correcting Codes
  6. Depolarizing Noise
  7. Cycle Structure of Quantum Tanner Codes
  8. Cycle Structure of HGP and LP Codes
  9. Proposed Decoder
  10. Trellis-Based MAP decoder
  11. Improved BP Decoding Using Generalized Checks
  12. Numerical Results
  13. Conclusion and Future Work
  14. Relay-BP$_4$

Key Result

Lemma 1

The incidence graph $I(\mathscr{G}^\square_i)$ is equal to the Tanner graph of $\mathscr{C}_i$ where, for each vertex in $\mathscr{G}^\square_i$, the $k_\textnormal{A}k_\textnormal{B}$ checks associated with that vertex are combined.

Figures (4)

  • Figure 1: Comparing the logical error rate performance of (generalized) MBP$_4$+OSD-$1$ decoding (with $\alpha=1.6$; $\alpha=1.5$ for the BB code) of three different $[[144,12]]$ qLDPC codes on the depolarizing channel. Solid curves are for standard MBP$_4$+OSD-$1$ decoding, while dashed and dash-dotted curves are for generalized MBP$_4$+OSD-$1$ decoding.
  • Figure 2: Comparing the logical error rate performance of (generalized) MBP$_4$ decoding with and without OSD-$1$ post-processing (with $\alpha=1.6$) of the $[[432,20,\leq 22]]$ quantum Tanner code from LeverrierRozendaalZemor25_1sub (dark blue curves), the $[[432,16,\leq 26]]$ quantum Tanner code from Wang-etal26_1sub (light blue curves), and the $[[416,18,\leq 22]]$ LP code and the $[[377,25,5]]$ HGP code from the GitHub qLDPC code repository Liu2026database (green and red curves) on the depolarizing channel. Solid curves are for standard MBP$_4$+OSD-$1$ decoding, dashed curves are for generalized MBP$_4$+OSD-$1$ decoding, while curves with dots are for decoding without OSD-$1$ post-processing (dotted curves for standard MBP$_4$ decoding with a maximum of $6$ BP iterations and dash-dotted curves for generalized MBP$_4$ decoding with a maximum of $25$ BP iterations).
  • Figure 3: Comparing the logical error rate performance of (generalized) MBP$_4$+OSD-$1$ decoding (with $\alpha=1.6$) of the $[[432,16,\leq 26]]$ quantum Tanner code from Wang-etal26_1sub for different values of $r$ on the depolarizing channel. The solid light blue curve is for standard MBP$_4$+OSD-$1$ decoding, and dashed to dotted light blue and red curves correspond (in this order) to increasing values of $r$ ($r=3,4,6,12$). As a comparison, the performance of Relay-BP$_4$ (Algorithm \ref{['alg:Relay-BP4']}) is shown with different values for the maximum number of relay legs ($R$ in Algorithm \ref{['alg:Relay-BP4']}) and the maximum number of BP iterations for each leg ($T_r$ in Algorithm \ref{['alg:Relay-BP4']}) (orange curves; the orange dotted curve has the same $R$ and $T_r$ as the dashed one, but with OSD-$1$ post-processing). The green curve is with no OSD-$1$ post-processing, but with a maximum of $50$ BP iterations.
  • Figure 4: Comparing the logical error rate performance of (generalized) MBP$_4$+OSD-$1$ decoding of two (generalized) quantum Tanner codes based on Cayley (dark and light blue curves) and Schreier (orange and magenta curves) graphs, respectively, and a GB code (brown and black curves) on the depolarizing channel for $n=686$ ($688$ for the GB code). Solid curves are for standard MBP$_4$+OSD-$1$ decoding, while dashed curves are for generalized MBP$_4$+OSD-$1$ decoding. For all three codes, two different values for $\alpha$ ($1.0$ and $1.6$) are compared.

Theorems & Definitions (14)

  • Definition 1: Incidence graph
  • Definition 2: Line graph
  • Definition 3: Tanner code
  • Definition 4: CSS code
  • Definition 5: Quadripartite quantum Tanner code
  • Definition 6: HGP code
  • Definition 7: LP code
  • Definition 8: $2$-TNC
  • Lemma 1
  • Proposition 1
  • ...and 4 more