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Construction and Decoding of Quantum Margulis Codes

Michele Pacenti, Dimitris Chytas, Bane Vasic

TL;DR

This work introduces quantum Margulis codes, a new QLDPC family derived from Margulis' classical LDPC construction via the 2BGA left-right Cayley complex. They show that these codes admit a standard min-sum decoder with complexity $O(n)$ under depolarizing noise, due to reduced Tanner graph symmetry which mitigates error degeneracy. A girth-control algorithm enables finite-length codes with girth 6 or 8, yielding instances at lengths $n=240$ and $n=642$ that outperform BB codes in the error-floor region with depolarizing noise, while remaining efficient to decode. Under circuit-level noise, however, the advantage diminishes and OSD post-processing becomes necessary; the paper emphasizes the need for new decoders that leverage the original Tanner-graph structure and symmetry properties for hardware-relevant noise models.

Abstract

Quantum low-density parity-check codes are a promising approach to fault-tolerant quantum computation, offering potential advantages in rate and decoding efficiency. In this work, we introduce quantum Margulis codes, a new class of QLDPC codes derived from Margulis' classical LDPC construction via the two-block group algebra framework. We show that quantum Margulis codes, unlike bivariate bicycle codes which require ordered statistics decoding for effective error correction, can be efficiently decoded using a standard min-sum decoder with linear complexity, when decoded under depolarizing noise. This is attributed to their Tanner graph structure, which does not exhibit group symmetry, thereby mitigating the well-known problem of error degeneracy in QLDPC decoding. To further enhance performance, we propose an algorithm for constructing 2BGA codes with controlled girth, ensuring a minimum girth of 6 or 8, and use it to generate several quantum Margulis codes of length 240 and 642. We validate our approach through numerical simulations, demonstrating that quantum Margulis codes behave significantly better than BB codes in the error floor region, under min-sum decoding.

Construction and Decoding of Quantum Margulis Codes

TL;DR

This work introduces quantum Margulis codes, a new QLDPC family derived from Margulis' classical LDPC construction via the 2BGA left-right Cayley complex. They show that these codes admit a standard min-sum decoder with complexity under depolarizing noise, due to reduced Tanner graph symmetry which mitigates error degeneracy. A girth-control algorithm enables finite-length codes with girth 6 or 8, yielding instances at lengths and that outperform BB codes in the error-floor region with depolarizing noise, while remaining efficient to decode. Under circuit-level noise, however, the advantage diminishes and OSD post-processing becomes necessary; the paper emphasizes the need for new decoders that leverage the original Tanner-graph structure and symmetry properties for hardware-relevant noise models.

Abstract

Quantum low-density parity-check codes are a promising approach to fault-tolerant quantum computation, offering potential advantages in rate and decoding efficiency. In this work, we introduce quantum Margulis codes, a new class of QLDPC codes derived from Margulis' classical LDPC construction via the two-block group algebra framework. We show that quantum Margulis codes, unlike bivariate bicycle codes which require ordered statistics decoding for effective error correction, can be efficiently decoded using a standard min-sum decoder with linear complexity, when decoded under depolarizing noise. This is attributed to their Tanner graph structure, which does not exhibit group symmetry, thereby mitigating the well-known problem of error degeneracy in QLDPC decoding. To further enhance performance, we propose an algorithm for constructing 2BGA codes with controlled girth, ensuring a minimum girth of 6 or 8, and use it to generate several quantum Margulis codes of length 240 and 642. We validate our approach through numerical simulations, demonstrating that quantum Margulis codes behave significantly better than BB codes in the error floor region, under min-sum decoding.

Paper Structure

This paper contains 13 sections, 3 theorems, 18 equations, 15 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. CSS codes, Tanner graphs, Cayley graphs
  4. Error model and decoding
  5. Two-block group-algebra code construction
  6. Automorphism group of 2BGA codes
  7. Quantum Margulis Codes
  8. Classical Margulis construction
  9. Quantum Margulis construction
  10. Constructing 2BGA codes with high girth
  11. Decoding of quantum Margulis code
  12. Performance of quantum Margulis codes
  13. Conclusions

Key Result

Theorem 1

Let $\mathcal{T} = (\mathcal{V} \cup \mathcal{C},\mathcal{E})$ be the $X$ (or $Z$) Tanner graph of a 2BGA code obtained from a group $G$ and two sets of generators $A,B$. Then, if $A$ ($B$) is a normal subgroup of $G$, $G \subseteq \mathrm{Aut}(\mathcal{T})$, assuming it to act on the right (left).

Figures (15)

  • Figure 1:
  • Figure 2:
  • Figure 4: 4-cycle in the expanded neighborhood of a check with $d_c=4$. Red, square nodes indicate checks, while circles indicate variable nodes. Black and gray variable nodes indicate qubits belonging to the two blocks $\mathbf{A}$ and $\mathbf{B}$.
  • Figure 5: 6-cycle in the expanded neighborhood of a check with $d_c=4$.
  • Figure 6: A symmetric stabilizer for a $d_v=3$ 2BGA code.
  • ...and 10 more figures

Theorems & Definitions (8)

  • Definition 1
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Proposition 1
  • proof
  • Definition 2