Heterogeneous Quantum Federated Learning

Updated 1 December 2025

Heterogeneous QFL frameworks formalize decentralized training of quantum or hybrid models by addressing diverse data distributions, device topologies, and noise characteristics.
They incorporate noise- and hardware-aware aggregation, layerwise model updates, and encoding-aware weighting to ensure robust, convergent learning in non-IID settings.
Empirical evaluations report enhanced accuracy and stability, with methods like Fisher-weighted aggregation and dynamic depth selection mitigating client variability.

Heterogeneous Quantum Federated Learning (QFL) frameworks formalize the decentralized training of parameterized quantum (or hybrid quantum–classical) models in distributed environments where clients differ substantially in data distribution, quantum hardware topology, qubit count, decoherence rates, quantum/classical processing capacity, and quantum encoding. Unlike classical FL, the quantum context introduces unique heterogeneities in quantum circuit depth, native gate sets, noise models, and even the mapping from classical data to quantum states. Addressing these variances is essential for robust, convergent, and high-performing collaborative quantum learning. The following sections systematically present the mathematical foundations, algorithmic techniques, architectures, convergence theory, empirical findings, and open research trajectories for heterogeneous QFL based on the latest literature.

1. Formalizing Heterogeneity in Quantum Federated Learning

In QFL, each client $i$ possesses a quantum dataset $\mathcal{D}_i$ , encoding function $E_i$ , and a parametrized quantum model (typically a variational quantum circuit; PQC) defined on $q_i$ qubits and with parameters $\omega_i \in \mathbb{R}^{d_i}$ . Heterogeneity arises in two main domains:

Data heterogeneity: Clients differ in $p_i(x)$ (local data distributions, including both raw data and label distributions) and in the encoding $E_i$ , so that even identical data instances yield non-orthogonal quantum states across clients.
System heterogeneity: Clients exhibit variation in device topology (e.g., connectivity graphs), qubit count, circuit depth (limited by coherence time and noise profile), and per-gate noise characteristics.

A global federated QFL objective is thus: $\min_{\omega} F(\omega) = \sum_{i=1}^N w_i F_i(\omega)$ where $F_i(\omega) = \mathbb{E}_{(x,y)\sim p_i}[\,L(f_i(x;\omega),y)\,]$ reflects all quantum and classical processing at client $i$ , and $w_i$ are aggregation weights, possibly encoding dataset size, device reliability, or noise compensation (Rahman et al., 27 Nov 2025).

The nonidentical parameter spaces ( $d_i$ ), encoding, and execution capability render standard parameter averaging approaches from classical FL both technically and theoretically questionable, necessitating new methodologies.

2. Algorithmic Approaches for Heterogeneous QFL

2.1 Noise- and Hardware-Aware Aggregation

Addressing noise and device variability, SpoQFL (Rahman et al., 15 Jul 2025), SPQFL (Rahman et al., 27 Nov 2025), and weighted protocols (Quy et al., 2 Nov 2024) employ dynamic per-client scaling based on local noise deviation or device characterization:

SpoQFL: Introduces a noise-adaptive scaling $x_n^t = \exp(-\gamma|\xi_n^t|)$ , where $\xi_n^t$ represents the estimated noise-induced gradient deviation at epoch $t$ , and only incorporates updates above a threshold $\tau$ . This mechanism down-weights or skips contributions from high-noise clients within each round.
Sporadic Participation: Clients failing to meet validation performance or noise-thresholded reliability can have their updates omitted entirely in aggregation.

2.2 Model and Depth Heterogeneity

The Quorus framework (Han et al., 30 Sep 2025) and related layerwise aggregation schemes (Rahman et al., 27 Nov 2025) allow clients to execute variational circuits at individually selected depths $d_i$ , driven by observed fidelity-loss in deeper circuits and hardware constraints. To mitigate objective mismatch, Quorus employs a layerwise loss, summing classification and inter-layer distillation losses across all intermediate circuit depths, so that parameter updates from shallow clients regularize the lower layers of deeper models.

2.3 Encoding and Data Heterogeneity

Multimodal and context-dependent QFL is addressed by modality-agnostic frameworks (Pokharel et al., 10 Jul 2025) using device- or modality-specific PQCs, fusion via entangling circuits, and gating off missing-modalities or noisier subcomponents through context vectors. Encoding-aware weighting or pre-alignment transformations adjust influence in the global model according to the Bures or trace distance between clients' input state distributions and a reference (Rahman et al., 27 Nov 2025).

2.4 Information-Theoretic and Knowledge-Preserving Fusion

QFedFisher (Bhatia et al., 23 Jul 2025) introduces Fisher information–based pruning: parameters with low empirical Fisher information on local data are averaged globally, while those deemed critical are preserved per-client using Fisher-weighted aggregation, thereby aligning updates with knowledge-rich local directions and improving non-IID robustness.

2.5 Scheduling, Hierarchical, and Asynchronous Protocols

Heterogeneous QFL in hierarchical and networked settings (e.g., LEO constellations, SAGIN) leverages role-aware scheduling (e.g., primary vs. secondary satellite partitioning in sat-QFL (Gurung et al., 20 Sep 2025)), adapts communication windows and aggregation order to time-varying connectivity, and incorporates direct inter-client communication (decentralized or two-tier) for load balancing (Rahman et al., 27 Nov 2025, Quy et al., 2 Nov 2024).

3. Mathematical Theory and Convergence

Convergence analyses for heterogeneous QFL account for client-specific noise variance ( $\eta_i$ ), data drift ( $\Delta_i$ ), and boundedness of model and system heterogeneities: $\mathbb{E}[F(\omega^K)] - F^* \leq (1-\eta\mu)^{KT}(F(\omega^0)-F^*) + O\Big( \frac{\eta \sum_i w_i \sigma_i^2}{\mu} + \frac{L^2 \eta \sum_i w_i \Delta_i^2 T}{\mu} \Big)$ for step-size $\eta \le 1/L$ , assuming $\mu$ -strong convexity, $L$ -smoothness, and bounded per-client variance and drift (Rahman et al., 27 Nov 2025). For nonconvex loss landscapes typical of quantum circuits, similar rates in the average gradient norm are established. Heterogeneity-aware schemes such as SpoQFL, QFedFisher, and Quorus have been demonstrated to empirically reduce the steady-state error introduced by client drift and quantum noise (Rahman et al., 15 Jul 2025, Bhatia et al., 23 Jul 2025, Han et al., 30 Sep 2025).

4. Architectural Elements and Design Patterns

Device and Protocol Taxonomy

Pure QFL: Each client trains a fully quantum variational model, expressively compressed by leveraging superposition and entanglement; subcircuit widths are dynamically selected per resource (Sai et al., 20 Oct 2025).
Hybrid QFL: Quantum circuits are appended to classical frontends, ensuring fallback to classical inference on low-capacity devices.
Centralized vs. Hierarchical vs. Decentralized Topologies: Centralized QFL employs a single aggregator; hierarchical QFL uses edge or cluster-level servers for intra-group aggregation; decentralized solutions leverage peer-to-peer quantum-secure graphs, with consensus via gossip or blockchain protocols (Sai et al., 20 Oct 2025, Quy et al., 2 Nov 2024).

Model and Data Processing Strategies

Heterogeneity Type	Algorithmic Remedy	Reference
Data encoding	Harmonization, state/encoding weights	(Rahman et al., 27 Nov 2025)
Model depth/width	Layerwise aggregation, slimmable QNNs	(Han et al., 30 Sep 2025, Sai et al., 20 Oct 2025)
Circuit topology	Qubit-aware embedding, layer-wise avg	(Rahman et al., 27 Nov 2025, Gurung et al., 2023)
Client noise	Noise-adaptive scaling, filtering	(Rahman et al., 15 Jul 2025, Rahman et al., 27 Nov 2025)
Multimodal fusion	Modal PQCs + entanglement fusion	(Pokharel et al., 10 Jul 2025)
Straggler effect	Asynchronous, staleness-aware updates	(Gurung et al., 20 Sep 2025, Quy et al., 2 Nov 2024)

Security primitives include quantum key distribution (QKD), quantum (homomorphic) encryption, differential privacy extended to quantum measurements, and blind (verifiable) quantum computing (Sai et al., 20 Oct 2025, Gurung et al., 20 Sep 2025).

5. Empirical Evaluation and Benchmarks

Recent works report the following empirical outcomes:

SpoQFL (Rahman et al., 15 Jul 2025): Outperforms conventional QFL by +4.87% (CIFAR-10) and +3.66% (CIFAR-100) accuracy while reducing cross-entropy loss, displaying robustness to Pauli noise and stability under heterogeneous error rates.
Quorus (Han et al., 30 Sep 2025): Achieves +12.4% average test accuracy gain over classical heterogeneous FL in PQC scenarios of varying depth; maintains non-vanishing gradient norms in shallow circuit layers, allowing deeper clients to refine parameter hierarchies.
QFedFisher (Bhatia et al., 23 Jul 2025): Integrating Fisher-weighted aggregation raises MNIST test accuracy to 91.2% (vs. 84.8% for FedAvg QFL) under Dirichlet non-IID splits, with modest additional compute overhead.
Multimodal QFL (Pokharel et al., 10 Jul 2025): Modality-agnostic, entanglement-fused architectures yield +6.84% (IID) and +7.25% (non-IID) absolute accuracy improvements, demonstrating stable training in the presence of incomplete modalities.

Simulations and real-device experiments (IBM QPUs) confirm that error-adaptive depth selection and single-shot, information-rich measurement schemes are essential for maintaining convergence and accuracy under realistic noise and bandwidth conditions (Han et al., 30 Sep 2025).

6. Future Research Directions and Open Challenges

Persistent open problems identified by the literature include:

Nonconvex convergence guarantees: Extending formal convergence bounds to general nonconvex QNN loss landscapes.
Model and architecture fusion: Aggregating heterogeneous quantum architectures, including different ansatz types, qubit connectivities, and gate sets.
Error mitigation and robust aggregation: Integrating on-device error correction/mitigation strategies and robust server-side filters to resist biased or adversarial updates.
Scalable, communication-efficient QFL: Designing protocols that sustain learning performance at scale with dynamic client participation and minimal overhead.
Quantum-classical hierarchical integration: Co-optimizing classical and quantum resources in multi-tier federated networks, particularly in 6G, SAGIN, and satellite constellations (Quy et al., 2 Nov 2024, Gurung et al., 20 Sep 2025).
Standardizing encodings and privacy: Developing benchmarks for data embedding and extending quantum-resistant cryptographic and privacy tools to distributed quantum ML.
Dynamic, resource-aware scheduling: Allowing clients to adapt circuit depth, model size, and participation in real-time as device calibrations and network conditions drift (Han et al., 30 Sep 2025, Rahman et al., 27 Nov 2025).