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Belief Propagation with Quantum Messages for Symmetric Q-ary Pure-State Channels

Avijit Mandal, Henry D. Pfister

TL;DR

This work extends belief propagation with quantum messages (BPQM) to symmetric $q$-ary pure-state channels by exploiting circulant Gram matrices. It derives closed-form eigenlist update rules for check- and bit-node combining, and provides explicit BPQM unitaries along with analytic fidelity bounds, enabling a density-evolution framework for polar and LDPC code design on these channels. The approach yields practical methods to estimate decoding performance and design parameters without large-scale quantum-state simulations, including polarCode design for target error rates and LDPC decoding thresholds. The results pave the way for efficient quantum-assisted decoding on CQ channels and suggest natural extensions to broader symmetry groups via group characters.

Abstract

Belief propagation with quantum messages (BPQM) provides a low-complexity alternative to collective measurements for communication over classical--quantum channels. Prior BPQM constructions and density-evolution (DE) analyses have focused on binary alphabets. Here, we generalize BPQM to symmetric q-ary pure-state channels (PSCs) whose output Gram matrix is circulant. For this class, we show that bit-node and check-node combining can be tracked efficiently via closed-form recursions on the Gram-matrix eigenvalues, independent of the particular physical realization of the output states. These recursions yield explicit BPQM unitaries and analytic bounds on the fidelities of the combined channels in terms of the input-channel fidelities. This provides a DE framework for symmetric q-ary PSCs that allows one to estimate BPQM decoding thresholds for LDPC codes and to construct polar codes on these channels.

Belief Propagation with Quantum Messages for Symmetric Q-ary Pure-State Channels

TL;DR

This work extends belief propagation with quantum messages (BPQM) to symmetric -ary pure-state channels by exploiting circulant Gram matrices. It derives closed-form eigenlist update rules for check- and bit-node combining, and provides explicit BPQM unitaries along with analytic fidelity bounds, enabling a density-evolution framework for polar and LDPC code design on these channels. The approach yields practical methods to estimate decoding performance and design parameters without large-scale quantum-state simulations, including polarCode design for target error rates and LDPC decoding thresholds. The results pave the way for efficient quantum-assisted decoding on CQ channels and suggest natural extensions to broader symmetry groups via group characters.

Abstract

Belief propagation with quantum messages (BPQM) provides a low-complexity alternative to collective measurements for communication over classical--quantum channels. Prior BPQM constructions and density-evolution (DE) analyses have focused on binary alphabets. Here, we generalize BPQM to symmetric q-ary pure-state channels (PSCs) whose output Gram matrix is circulant. For this class, we show that bit-node and check-node combining can be tracked efficiently via closed-form recursions on the Gram-matrix eigenvalues, independent of the particular physical realization of the output states. These recursions yield explicit BPQM unitaries and analytic bounds on the fidelities of the combined channels in terms of the input-channel fidelities. This provides a DE framework for symmetric q-ary PSCs that allows one to estimate BPQM decoding thresholds for LDPC codes and to construct polar codes on these channels.
Paper Structure (39 sections, 17 theorems, 134 equations, 7 figures, 4 algorithms)

This paper contains 39 sections, 17 theorems, 134 equations, 7 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Notation
  4. Preliminaries
  5. Symmetric $q$-ary Pure State Channel
  6. Information Measures
  7. Pretty Good Measurement on Symmetric $q$-ary PSC
  8. BPQM on Symmetric PSC
  9. Check node and Bit node Unitary
  10. Fidelity Bounds
  11. Density Evolution
  12. Polar Codes
  13. LDPC Codes
  14. Conclusion
  15. Quantum Preliminaries
  16. ...and 24 more sections

Key Result

Lemma 5

For $m\in[q]$, $\ket{v_{m}}$ is an eigenvector of $G$ satisfying $G \ket{v_m} = \lambda_m \ket{v_m}$ where $\lambda_{m}=\sum_{j\in [q]}g_{j}\omega^{jm}$$\forall m\in [q]$.

Figures (7)

  • Figure 1: Plot comparing rate $\frac{|\mathcal{A}|}{N}$ with $N=2^n$ of polar codes for $q=3$
  • Figure 2: Threshold plot for $(3,6)$ LDPC code with $q=3$
  • Figure 3: Polar design curves for $q=3$ and $q=5$ for different block lengths $N=2^n$
  • Figure 4: Polar design curves for $q=3$ and $q=5$ for different block lengths $N=2^n$
  • Figure 5: Plot comparing rate $\frac{|\mathcal{A}|}{N}$ of polar codes with different block lengths
  • ...and 2 more figures

Theorems & Definitions (42)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 5
  • Lemma 6
  • proof
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • ...and 32 more