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Quantum Approximate Optimization for Decoding of Low-Density Parity-Check Codes

Krishnakanta Barik, Goutam Paul

TL;DR

A QAOA-based decoding framework for LDPC codes is presented by formulating a decoding cost function that incorporates both parity-check constraints and soft channel reliability information and demonstrates that the QAOA-based decoder achieves a higher probability of correctly recovering the transmitted codeword than BP across multiple experimental settings.

Abstract

Decoding Low-Density Parity-Check (LDPC) codes is a fundamental problem in coding theory, and Belief Propagation (BP) is one of the most popular methods for LDPC code decoding. However, BP may encounter convergence issues and suboptimal performance, especially for short-length codes and in high-noise channels. The Quantum Approximate Optimization Algorithm (QAOA) is a type of Variational Quantum Algorithm (VQA) designed to solve combinatorial optimization problems by minimizing a problem-specific cost function. In this paper, we present a QAOA-based decoding framework for LDPC codes by formulating a decoding cost function that incorporates both parity-check constraints and soft channel reliability information. The resulting optimization problem is solved using QAOA to search for low-energy configurations corresponding to valid codewords. We test the proposed method through extensive numerical experiments and compare its performance with BP decoding. The experimental results demonstrate that the QAOA-based decoder achieves a higher probability of correctly recovering the transmitted codeword than BP across multiple experimental settings.

Quantum Approximate Optimization for Decoding of Low-Density Parity-Check Codes

TL;DR

A QAOA-based decoding framework for LDPC codes is presented by formulating a decoding cost function that incorporates both parity-check constraints and soft channel reliability information and demonstrates that the QAOA-based decoder achieves a higher probability of correctly recovering the transmitted codeword than BP across multiple experimental settings.

Abstract

Decoding Low-Density Parity-Check (LDPC) codes is a fundamental problem in coding theory, and Belief Propagation (BP) is one of the most popular methods for LDPC code decoding. However, BP may encounter convergence issues and suboptimal performance, especially for short-length codes and in high-noise channels. The Quantum Approximate Optimization Algorithm (QAOA) is a type of Variational Quantum Algorithm (VQA) designed to solve combinatorial optimization problems by minimizing a problem-specific cost function. In this paper, we present a QAOA-based decoding framework for LDPC codes by formulating a decoding cost function that incorporates both parity-check constraints and soft channel reliability information. The resulting optimization problem is solved using QAOA to search for low-energy configurations corresponding to valid codewords. We test the proposed method through extensive numerical experiments and compare its performance with BP decoding. The experimental results demonstrate that the QAOA-based decoder achieves a higher probability of correctly recovering the transmitted codeword than BP across multiple experimental settings.
Paper Structure (10 sections, 1 theorem, 30 equations, 1 figure, 1 table)

This paper contains 10 sections, 1 theorem, 30 equations, 1 figure, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Classical Preliminaries
  4. Quantum Preliminaries
  5. QAOA for LDPC Decoding
  6. Experimental Results
  7. Experimental Configuration
  8. Results and Discussion
  9. Advantages
  10. Conclusion and Future Directions

Key Result

Theorem 1

Let $H(z) = H_{\mathrm{parity}}(z) + H_{\mathrm{channel}}(z)$ be the decoding cost Hamiltonian defined as where $Z_i = (-1)^{x_i}$ with $x_i \in \{0,1\}$, and $h_i$ denotes the channel Log-Likelihood Ratio associated with the $i$-th bit. Then any minimizer of $H(Z)$ corresponds to a codeword that satisfies all parity-check constraints and is maximally consistent with the channel observations.

Figures (1)

  • Figure 1: Energy convergence of the QAOA-based decoder versus optimization steps for three different experimental settings ($\sigma = 1, 1.5, 2$).

Theorems & Definitions (6)

  • Definition 1: Code richardson2008modern
  • Definition 2: Linear Code richardson2008modern
  • Definition 3: Parity-Check Matrix richardson2008modern
  • Definition 4: Tanner Graph richardson2008modern
  • Definition 5: Low-Density Parity-Check Codes richardson2008modern
  • Theorem 1