Quantum Message Passing for Factor Graphs over Finite Abelian Groups

Abstract: We develop a quantum message-passing framework for factor graphs over finite abelian groups. Our starting point is the task of discriminating between a collection of quantum states indexed by the elements of a finite abelian group $\mathcal{G}$ whose overlaps respect the structure of a group-covariant pure-state channel (PSC). For such channels, we show that the Gram matrix constructed from the output states is diagonalized by the character basis of the dual group $\widehat{\mathcal{G}}$. Hence, the channel is characterized, up to isometric equivalence, by its character-indexed eigen list. Based on this representation, we analyze the induced classical-quantum channels associated with check, equality, homomorphism, marginalization, and automorphism factors. For each factor, we derive explicit update rules showing that if the incoming messages are heralded mixtures of group-covariant PSCs, then the outgoing message remains in the same class. This provides a closed quantum message-passing framework for tree-structured factor graphs assembled from these primitives. The framework applies directly to several standard code families over finite abelian groups, including polar codes, LDPC codes, and convolutional and turbo codes. It recovers the previously studied $q$-ary formulation as the special case $(\mathcal{G}=\mathbb{Z}_q)$, while extending the belief propagation with quantum messages (BPQM) framework introduced by Renes to non-cyclic alphabets and more general factor-graph constraints described by homomorphisms between products of abelian groups.

• The paper introduces a closed-form quantum message passing framework that generalizes belief propagation to finite abelian groups using an eigen-based parametrization.
• It derives explicit update rules for check, equality, homomorphism, marginalization, and automorphism factors, linking quantum inference with group representation theory.
• The framework supports density evolution analysis for various coding schemes, demonstrating near-optimal thresholds and scalable computational performance.

Quantum Message Passing for Factor Graphs over Finite Abelian Groups

Introduction

This paper develops a comprehensive quantum message-passing framework for factor graphs defined over finite abelian groups, establishing both the operational structure and theoretical foundation for efficient decoding in quantum-enabled communication and inference settings. The work systematically extends belief propagation with quantum messages (BPQM) beyond previous binary and $q$-ary results to arbitrary finite abelian alphabets, leveraging group-theoretic structure to produce a closed-form, eigen-based formalism for quantum inference on locally constrained graphical models. The approach connects quantum information theory, group representation theory, and modern coding/inference architectures (such as LDPC, turbo, and polar codes), and provides a rigorous path for density evolution (DE) and threshold analysis of quantum decoders.

Group-Covariant Quantum Channels and Factor Graph Structure

The framework is built upon quantum pure-state channels (PSCs) that are covariant under the action of a finite abelian group $G$. The key structural insight is that the Gram matrix, representing pairwise overlaps of the channel’s pure outputs $\{\ket{\psi_g}\}_{g \in G}$, is always $G$-circulant and is diagonalized by the character basis of the dual group $\widehat{G}$—each eigenvalue corresponding to a character uniquely determines the channel up to isometric equivalence. This observation underpins the “eigen list” parametrization of CQ channels, permitting efficient propagation and update of sufficient quantum statistics along factor graphs.

Each variable in the factor graph takes values in $G$, and local constraints (equality, parity, general homomorphisms, automorphisms, marginalizations) are encoded by appropriate factors. Notably, these primitives subsume all local operations needed for classical code architectures including polar, LDPC, group codes, and trellis-based (convolutional/turbo) codes.

Quantum Message-Passing Update Rules

The main technical contribution is the derivation of closed-form quantum message-passing update rules for each local factor type, specifying how eigen list statistics and classical side information (“herald registers”) are recursively transformed:

• Check (parity) factors: Combine two $G$-covariant PSCs and produce an explicit heralded mixture of output PSCs, each characterized by a transformed eigen list parameterized by a character of $G$.
• Equality factors: The output remains a single $G$-covariant PSC, with an eigen list given by a convolution of the inputs in the character domain; no new herald arises.
• Homomorphism factors: For a surjective homomorphism $\phi: G_1 \to G_2$, the output is a heralded mixture indexed by coset representatives in $G$0; each mixture component carries an eigen list mapped via the dual homomorphism. If the input eigen list is supported only on $G$1, the output is a single PSC.
• Marginalization: Projection onto a product group coordinate leads to a heralded mixture, with the discarded dual coordinate serving as the herald.
• Automorphism factors: The output eigen list is simply permuted according to the dual automorphism.

These updates are described both operationally (as explicit isometries/unary operations) and algebraically in terms of group characters and Fourier transform indices, enabling algorithmic implementation.

Figure 1: DE Threshold Curve for Turbo Codes with each constituent convolutional decoder $G$2.

Figure 2: Two-Dimensional Heatmap for DE Success Probability for Turbo Code with each constituent convolutional decoder $G$3.

Closure and Universality on Tree-Structured Factor Graphs

A central result establishes that, for any tree-structured factor graph composed solely of the above primitives, quantum message passing is closed under heralded mixtures of group-covariant PSCs. This means that all messages throughout the computation tree can be succinctly and completely described by finite ensembles of eigen lists with explicit classical mixing (induced by local symmetry-breaking or marginalization events), preserving tractability and enabling efficient recursion on arbitrary codes defined over finite abelian alphabets.

Applications to Code Families

The framework unifies and extends quantum decoding architectures:

• Polar codes: Kernel operations decomposed into automorphism, check, and equality factors allow recursive tracking of synthetic channels via character-indexed eigen lists for general finite abelian groups, subsuming non-cyclic and field-based constructions.
• LDPC/group codes: Tanner graphs specified by equality, parity, and automorphism factors directly admit local quantum message-passing rules; maximal-likelihood inference is achieved on trees.
• Convolutional and turbo codes: The constituent trellis section is described via surjective homomorphism and marginalization updates, with the forward recursion producing symbol- and state-wise MAP messages as heralded mixtures of eigen lists. Turbo decoding is constructed by coupling two such trellis structures, with interleaving corresponding to automorphism relabeling and equality constraints enforced by the equality update.

This approach captures and extends previously established binary and $G$4-ary BPQM schemes, realizing them as special or degenerate cases of the more general group-theoretic machinery developed herein.

Density Evolution and Performance Analysis

With the eigen list parametrization as the sufficient statistic, density evolution (DE) analyses analogous to their classical and $G$5-ary quantum counterparts are enabled generically. DE carries out asymptotic tracking of the message population under recursive updates, estimating thresholds and symbol error rates for various quantum decoders. The paper leverages information-theoretic metrics (symmetric Holevo information, channel fidelity, PGM error rates) that are directly computable from the local/posterior eigen lists at each recursion step.

Strong numerical results are presented for turbo code ensembles, notably:

• As seen in (Figure 1), the DE threshold for a rate-$G$6 turbo code with $G$7 is $G$8, while the corresponding Holevo threshold is $G$9, demonstrating that the quantum message-passing decoder exhibits a near-optimal gap to the ultimate capacity.
• The two-dimensional success probability heatmaps (Figure 2) exhibit the sharpness and geometry of the achievable region in the parameter space of channel eigen lists, with DE and Holevo boundaries indicated.

Theoretical and Practical Implications

The group-theoretic structure and closure under message-passing recursions constitute a unified, algebraic foundation for quantum inference on structured code ensembles, enabling efficient algorithmic instantiations as well as analytic study (e.g., tracking of performance thresholds, optimal symbol discrimination). The use of the character basis as a canonical message domain means that symmetries of the problem are inherently captured, minimizing the memory overhead and computational burden (especially relative to basis-dependent or state-ensemble approaches).

The framework further opens the door to:

• Systematic design of quantum decoders for non-cyclic, composite, or product group alphabets (prominent in practical quantum and classical systems)
• Extension of DE and threshold analysis to arbitrary group-based code families, supporting cross-family performance comparison and code optimization
• Incorporation of more general (possibly noisy or mixed) quantum channels, since the eigen list algebra is robust to statistical mixture operations
• Application to quantum machine learning on factor graphs, where local constraints often obey algebraic symmetries

Conclusion

This work provides a rigorous and unified abelian-group extension of quantum message passing and BPQM for factor-graph-based decoding and inference. By identifying the dual-group character domain as the natural representation for quantum messages and factoring the problem in terms of eigen lists and heralded mixtures, the framework achieves closure and tractability on tree-structured graphs, and delivers explicit algorithmic procedures for a broad class of error correction codes and inference problems. The formalism supports exact local recursion, principled performance tracking (via DE), and facilitates future quantum code and algorithm design in both theoretical and practical contexts.

Reference: "Quantum Message Passing for Factor Graphs over Finite Abelian Groups" (2604.12186)