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Breaking the Orthogonality Barrier in Quantum LDPC Codes

Kenta Kasai

TL;DR

The paper addresses the challenge of constructing quantum LDPC (QLDPC) codes that must satisfy orthogonality between CSS parity-check matrices, a constraint that typically degrades girth and imposes distance upper bounds. It introduces a design principle based on generalized Hagiwara–Imai codes with block-circulant structure and affine permutation matrices (APMs) to realize active orthogonality $H_XH_Z^{\mathsf T}=0$ while allowing latent non-orthogonality, thereby decoupling orthogonality, regularity, girth, and distance. A concrete $(3,12)$-regular, girth-8 $[[9216,4612,\le 48]]$ code is constructed with $P=768$, achieving FER $10^{-8}$ on a depolarizing channel with $p=4\%$ under BP decoding plus post-processing, and the authors develop an ETS-based post-processing framework to suppress harmful trapping structures. The work provides a practical construction pathway that preserves classical LDPC parity-check structure for CSS codes, demonstrates improved distance behavior, and offers decoding techniques (FHD, OSD, ETS) that yield strong finite-length performance with potential scalability to larger quantum codes.

Abstract

Classical low-density parity-check (LDPC) codes are a widely deployed and well-established technology, forming the backbone of modern communication and storage systems. It is well known that, in this classical setting, increasing the girth of the Tanner graph while maintaining regular degree distributions leads simultaneously to good belief-propagation (BP) decoding performance and large minimum distance. In the quantum setting, however, this principle does not directly apply because quantum LDPC codes must satisfy additional orthogonality constraints between their parity-check matrices. When one enforces both orthogonality and regularity in a straightforward manner, the girth is typically reduced and the minimum distance becomes structurally upper bounded. In this work, we overcome this limitation by using permutation matrices with controlled commutativity and by restricting the orthogonality constraints to only the necessary parts of the construction, while preserving regular check-matrix structures. This design breaks the conventional trade-off between orthogonality, regularity, girth, and minimum distance, allowing us to construct quantum LDPC codes with large girth and without the usual distance upper bounds. As a concrete demonstration, we construct a girth-8, (3,12)-regular $[[9216,4612, \leq 48]]$ quantum LDPC code and show that, under BP decoding combined with a low-complexity post-processing algorithm, it achieves a frame error rate as low as $10^{-8}$ on the depolarizing channel with error probability $4 \%$.

Breaking the Orthogonality Barrier in Quantum LDPC Codes

TL;DR

The paper addresses the challenge of constructing quantum LDPC (QLDPC) codes that must satisfy orthogonality between CSS parity-check matrices, a constraint that typically degrades girth and imposes distance upper bounds. It introduces a design principle based on generalized Hagiwara–Imai codes with block-circulant structure and affine permutation matrices (APMs) to realize active orthogonality while allowing latent non-orthogonality, thereby decoupling orthogonality, regularity, girth, and distance. A concrete -regular, girth-8 code is constructed with , achieving FER on a depolarizing channel with under BP decoding plus post-processing, and the authors develop an ETS-based post-processing framework to suppress harmful trapping structures. The work provides a practical construction pathway that preserves classical LDPC parity-check structure for CSS codes, demonstrates improved distance behavior, and offers decoding techniques (FHD, OSD, ETS) that yield strong finite-length performance with potential scalability to larger quantum codes.

Abstract

Classical low-density parity-check (LDPC) codes are a widely deployed and well-established technology, forming the backbone of modern communication and storage systems. It is well known that, in this classical setting, increasing the girth of the Tanner graph while maintaining regular degree distributions leads simultaneously to good belief-propagation (BP) decoding performance and large minimum distance. In the quantum setting, however, this principle does not directly apply because quantum LDPC codes must satisfy additional orthogonality constraints between their parity-check matrices. When one enforces both orthogonality and regularity in a straightforward manner, the girth is typically reduced and the minimum distance becomes structurally upper bounded. In this work, we overcome this limitation by using permutation matrices with controlled commutativity and by restricting the orthogonality constraints to only the necessary parts of the construction, while preserving regular check-matrix structures. This design breaks the conventional trade-off between orthogonality, regularity, girth, and minimum distance, allowing us to construct quantum LDPC codes with large girth and without the usual distance upper bounds. As a concrete demonstration, we construct a girth-8, (3,12)-regular quantum LDPC code and show that, under BP decoding combined with a low-complexity post-processing algorithm, it achieves a frame error rate as low as on the depolarizing channel with error probability .
Paper Structure (18 sections, 4 theorems, 32 equations, 3 figures, 1 table)

This paper contains 18 sections, 4 theorems, 32 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem setting: low-weight logical operators induced by check-row removal
  3. Method of matrix design
  4. Sufficient condition for active orthogonality
  5. Necessary condition for latent non-orthogonality
  6. Upper bound on girth
  7. Method of Matrix Construction
  8. Method of Decoding
  9. Flip history decoding
  10. OSD
  11. Elementary Trapping Set (ETS)-based post-processing
  12. Results
  13. Structure and active orthogonality
  14. Design goal and commutation table
  15. Affine permutation construction and choice of $P$
  16. ...and 3 more sections

Key Result

Theorem 3.1

If $F_u$ and $G_{r-u}$ commute for all $r\in\Delta$ and all $u\in[L/2]$, then $H_XH_Z^{\mathsf T}=0$.

Figures (3)

  • Figure 1: ETSs formed by connecting $k=2,5$ length-8 cycles. Filled check nodes have degree 1 (unsatisfied). Counts for $P=768,J=3,L=12$: $(6,2)$ ETS (X=48, Z=16) and $(12,2)$ ETS (X=23, Z=0).
  • Figure 2: Tanner graph of an ETS with (8,2). The internal subgraph matches the $(6,2)$ ETS in Fig. \ref{['fig:ets-8cycles']}(a), and the length-4 path connects its former degree-1 checks (C0:599 and C0:542). Count for $P=768,J=3,L=12$: $(8,2)$ ETS (X=48, Z=0).
  • Figure 3: FER curve of the constructed code girth-8, (3,12)-regular $[[9216,4612, \leq 48]]$, ($P=768,J=3,L=12$) with error bars (95% confidence). Here $n=PL$, the active matrix ranks are $\mathrm{rank}(H_X)=2302$ and $\mathrm{rank}(H_Z)=2302$, hence $k=4612$.

Theorems & Definitions (8)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof