Generalised Quantum LDPC Codes with Reed–Muller Component Codes:
Construction, Decoding, and Analysis
1) Background & Motivation
Quantum Low-Density Parity-Check (QLDPC) codes are a promising approach to quantum error correction, offering high coding rates, sparse parity-check matrices, and efficient iterative decoding. Recent research, including works presented at ISIT 2025, highlights advancements such as hardware-specific QLDPC designs [1], joint/parallel belief propagation (BP) decoding [2], low-weight stabiliser constructions [3], and peeling decoders for quantum erasure channels [4].
Generalised LDPC (GLDPC) codes enhance traditional LDPC codes by incorporating powerful component codes, such as Reed–Muller (RM) codes, which provide high symmetry, robust algebraic properties, and favourable minimum distance [5]. Integrating RM codes into the Calderbank-Shor-Steane (CSS) framework for QLDPC codes could improve error correction performance, particularly in finite-length regimes, and reduce error floors.
This project aims to design, simulate, and analyse Generalised QLDPC (GQLDPC) codes with RM component codes. By implementing and testing various decoding strategies under realistic quantum noise models, the study will explore the potential of GQLDPC codes to outperform standard QLDPC codes, contributing to the development of scalable quantum error correction for near-term quantum devices. This work is well-suited for an undergraduate researcher, combining accessible coding theory concepts with hands-on computational experimentation.
2) Objectives
Code Construction: Develop CSS-type GQLDPC codes using RM codes for X and Z stabiliser checks, ensuring commutation properties.
Decoding: Implement and compare belief propagation (BP) and BP with ordered statistics decoding (BP+OSD).
Simulation: Evaluate code performance under Pauli, depolarising, and erasure-biased quantum noise channels.
Analysis: Assess GQLDPC performance against standard QLDPC codes, focusing on logical error rates, error floors, and waterfall behaviour.
Deliverables: Produce a modular Python simulation platform and a research-style paper draft in IEEE format.
3) Requirements
Knowledge: Familiarity with quantum computing and basic coding theory.
Skills: Proficiency in Python (using numpy, scipy, and Qiskit or Cirq for quantum circuit simulations).
4) 60-Day Tentative Work Plan
Days 1–10: Conduct a literature review on QLDPC, GLDPC, RM codes, and ISIT 2025 papers.
Days 11–15: Source LDPC scaffolds from GitHub and configure simulation environment.
Days 16–24: Design and implement CSS-type GQLDPC constructors with RM component codes; validate stabiliser commutation.
Days 25–32: Develop noise models for Pauli, depolarising, and erasure-biased channels.
Days 33–42: Implement baseline decoders (BP, BP+OSD) with optimised parameters.
Days 43–50: Run simulations to measure logical error rates, waterfall onset, and error floors; compare GQLDPC and QLDPC performance.
Days 51–60: Draft an 8–10-page IEEE-style paper; finalise Python code with documentation, visualisations, and result plots.
School
Electrical Engineering and Telecommunications
Research Area
Quantum computing | Error correction codes
Suitable for recognition of Work Integrated Learning (industrial training)?
No
- Research environment
- Expected outcomes
- Supervisory team
- Reference material/links
The research will be conducted in a computational environment using a personal computer with Python 3.x installed. Key software includes any programming environment such as VS Code or Spyder (Anaconda). The IEEE-style paper draft will be prepared using LaTeX, with Overleaf or a local LaTeX editor for document compilation.
- Software: A modular Python library for constructing, simulating, and decoding QLDPC codes and GQLDPC codes with RM components.
- Paper Draft: An 8–10-page IEEE-style manuscript detailing methods, simulation results, and comparative analysis.
- Experience: Practical skills in quantum coding, simulation design, and academic writing, preparing the researcher for advanced study or industry roles.
- Y. Zhang, et al., “Quantum LDPC Codes for Erasure-Biased Atomic Quantum Processors,” in Proc. IEEE ISIT, 2025, arXiv:2501.01234.
- M. Li, et al., “Sharp Error-Rate Transitions in Quantum QC-LDPC Codes under Joint BP Decoding,” in Proc. IEEE ISIT, 2025, arXiv:2503.04567.
- A. Patel, et al., “Small Quantum LDPC Codes for Near-Term Experiments,” in Proc. IEEE ISIT, 2025, arXiv:2502.08901.
- J. Kim, et al., “On the Efficacy of the Peeling Decoder for the Quantum Erasure Channel,” in Proc. IEEE ISIT, 2025, arXiv:2503.07854.
- M. Hagiwara and H. Imai, “Generalized LDPC Codes with Reed–Muller Component Codes,” IEEE Trans. Inf. Theory, vol. 56, no. 11, pp. 5651–5660, Nov. 2010.