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Polar Codes for CQ Channels: Decoding via Belief-Propagation with Quantum Messages

Avijit Mandal, S. Brandsen, Henry D. Pfister

TL;DR

Problem: design and decoding of polar codes for general CQ channels. Approach: PM-BPQM decoding with a DE framework to compute effective-channel reliabilities via the recursions $W_N^{(2i-1)}=W^{(i)}_{N/2}\boxast W^{(i)}_{N/2}$ and $W_N^{(2i)}=W^{(i)}_{N/2}\varoast W^{(i)}_{N/2}$, aided by Monte Carlo parameter tracking; information-set selection via smallest estimated Helstrom error. Contributions: DE-based polar-code design for CQ channels under PM-BPQM, a Python implementation of the PM-BPQM polar decoder, and simulations showing higher rates than hard-decision baselines with observable polarization. Significance: provides a practical route to rate–reliability trade-offs for CQ channels and demonstrates the viability of PM-BPQM-based polar decoding both theoretically and in software.

Abstract

This paper considers the design and decoding of polar codes for general classical-quantum (CQ) channels. It focuses on decoding via belief-propagation with quantum messages (BPQM) and, in particular, the idea of paired-measurement BPQM (PM-BPQM) decoding. Since the PM-BPQM decoder admits a classical density evolution (DE) analysis, one can use DE to design a polar code for any CQ channel and then efficiently compute the trade-off between code rate and error probability. We have also implemented and tested a classical simulation of our PM-BPQM decoder for polar codes. While the decoder can be implemented efficiently on a quantum computer, simulating the decoder on a classical computer actually has exponential complexity. Thus, simulation results for the decoder are somewhat limited and are included primarily to validate our theoretical results.

Polar Codes for CQ Channels: Decoding via Belief-Propagation with Quantum Messages

TL;DR

Problem: design and decoding of polar codes for general CQ channels. Approach: PM-BPQM decoding with a DE framework to compute effective-channel reliabilities via the recursions and , aided by Monte Carlo parameter tracking; information-set selection via smallest estimated Helstrom error. Contributions: DE-based polar-code design for CQ channels under PM-BPQM, a Python implementation of the PM-BPQM polar decoder, and simulations showing higher rates than hard-decision baselines with observable polarization. Significance: provides a practical route to rate–reliability trade-offs for CQ channels and demonstrates the viability of PM-BPQM-based polar decoding both theoretically and in software.

Abstract

This paper considers the design and decoding of polar codes for general classical-quantum (CQ) channels. It focuses on decoding via belief-propagation with quantum messages (BPQM) and, in particular, the idea of paired-measurement BPQM (PM-BPQM) decoding. Since the PM-BPQM decoder admits a classical density evolution (DE) analysis, one can use DE to design a polar code for any CQ channel and then efficiently compute the trade-off between code rate and error probability. We have also implemented and tested a classical simulation of our PM-BPQM decoder for polar codes. While the decoder can be implemented efficiently on a quantum computer, simulating the decoder on a classical computer actually has exponential complexity. Thus, simulation results for the decoder are somewhat limited and are included primarily to validate our theoretical results.
Paper Structure (26 sections, 6 theorems, 55 equations, 12 figures, 3 algorithms)

This paper contains 26 sections, 6 theorems, 55 equations, 12 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminaries
  4. Paired-Measurement BPQM
  5. Density Evolution for Paired-Measurement BPQM
  6. Polar Codes
  7. PM-BPQM and Polar Codes
  8. PM-BPQM DE for Polar Code Design
  9. Numerical Results for Polar Code Design
  10. Simulation of the PM-BPQM Polar Decoder
  11. Conclusion
  12. Qubit BSCQ Capacity Comparison
  13. Polarization of PM-BPQM Decoding
  14. Deferred Measurement Polar Decoding
  15. Length-4 PM-BPQM Polar Decoder
  16. ...and 11 more sections

Key Result

Lemma 2

Any BSCQ channel that outputs a qubit is unitarily equivalent to the qubit channel $W:\left\{ 0,1\right\} \to\mathcal{D}(\mathcal{H}_{2})$ satisfying $W(z)=\sigma_{x}^{z}\rho(\delta,\gamma)\sigma_{x}^{z}$ with for some $\delta\in[0,1]$ and $\gamma\in\mathbb{C}$ satisfying $\left|\gamma\right|^{2}\leq\delta(1-\delta)$.

Figures (12)

  • Figure 1: Comparison of PM-BPQM and MF polar decoder for $N=1024$ on qubit BSCQ channels with $\delta=0.07$ and variable $\gamma$ under a union-bound block-error constraint of 0.1.
  • Figure 2: Comparison of PM-BPQM and MF polar decoder with $N=1024$ on qubit BSCQ channels with $\delta=0.09$ and variable $\gamma$ under a union-bound block-error constraint of 0.1.
  • Figure 3: Comparison of bit-error rate between DE analysis (solid lines) and simulated decoder (circles) for the effective channels of bits $u_1,\ldots,u_8$ of a length-8 polar code over qubit BSCQ channels with $\delta=0.1$ and variable $\gamma$.
  • Figure 4: DE polar design curve for PM-BPQM decoding of a length-$2^{n}$ code on the qubit channel with $(\delta,\gamma)=(0.08,0.05)$.
  • Figure 5: Decoding factor graph for a length-4 polar code
  • ...and 7 more figures

Theorems & Definitions (14)

  • Definition 1
  • Lemma 2: brandsen_bpqm_arxiv
  • Definition 3
  • Definition 4
  • Lemma 5: brandsen2022belief
  • Remark 6
  • Lemma 7: brandsen2022belief
  • Lemma 8: brandsen2022belief
  • Lemma 9
  • Remark 10
  • ...and 4 more