A Stable Aggregation Method for Quantum Federated Learning

Shanika Nanayakkara; Shiva Raj Pokhrel

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A Stable Aggregation Method for Quantum Federated Learning

Shanika Nanayakkara, Shiva Raj Pokhrel

arXiv:2609.00356v1Jun 25, 2026cs.AIcs.LG

TL;DR

QFL aggregation must address heterogeneous client reliability and periodic QNN parameter geometry. The paper proposes SCM-A2G-QFL, combining QoS-aware weighting, torus-consistent directions, and midpoint self-consistency. The resulting update is normalized, geometry-preserving, locally stable under small geometry gains, and implicitly damped relative to direct A2G movement, with IBM hardware validation of executable angles and expected observable behavior.

Problem
QFL aggregation must jointly handle varying client reliability and periodic QNN parameter geometry, because Euclidean averaging can fail near angular seams.
Method
SCM-A2G-QFL assigns QoS-aware client weights, constructs a torus-consistent candidate direction, and accepts movement only when supported by its own midpoint.
Results
SCM-A2G-QFL produces normalized, geometry-preserving, locally stable, and implicitly damped updates, while IBM validation demonstrates executable angles and expected quantum observable behaviour.
Takeaways & Limitations
The framework provides a self-consistent aggregation approach for QFL that addresses reliability variation and periodic parameter structure within the evaluated scope.

Abstract

from arXiv · show

Quantum federated learning (QFL) enables clients to train quantum neural network (QNN) models without sharing private data. We find that aggregation in QFL is unstable under heterogeneous data, unreliable communication, variable fidelity, latency, and quantum hardware noise. Moreover, QFL is non-trivially challenging because several QNN parameters are periodic angles, where Euclidean averaging often fails to capture the inherent dynamics. We develop a novel self-consistent midpoint aggregation method for stable QFL design and implementation. We combine QoS-aware client weighting, circular parameter aggregation, and bounded midpoint-based update control. We perform several angular tests and IBM real Quantum machines experiments for validation confirming our approach. Extensive evaluations and experiments on medical and financial datasets show improved stability, lower volatility, and competitive accuracy.

I. INTRODUCTION

QFL aggregation must handle both heterogeneous client reliability and the periodic geometry of QNN rotation angles. The paper develops QoS-aware, circular, midpoint-controlled updates to improve stability and preserve geometry.

QFL aggregation is affected by non-IID data, unequal client resources, communication conditions, device instability, and quantum noise.
Periodic QNN rotation angles make Euclidean averaging unreliable near the −π/π seam, where geometrically adjacent values can appear far apart.
A. QFL Aggregation Problem and Ideas: 14.35° is the Euclidean mean for seam-adjacent angles, whereas the circular mean remains near 176.4°, consistent with the client cluster.
A. QFL Aggregation Problem and Ideas: A2G combines client reliability and update-control gains, while Circular A2G uses circular geometry for angular parameters.
A. QFL Aggregation Problem and Ideas: MP-A2G applies a one-shot midpoint correction to reduce overly aggressive movement toward an unstable aggregate.
A. QFL Aggregation Problem and Ideas: A2G, MP-A2G, and SCM-A2G avoid the misleading Euclidean aggregate and move conservatively toward a geometry-consistent angular region.

C. Key Contributions

The paper positions SCM-A2G-QFL as a quantum federated learning framework that jointly addresses unreliable client updates and periodic QNN parameter geometry. It combines QoS-aware weighting, geometry-aware aggregation, and midpoint-based control of the accepted server movement.

Key Contributions: The paper evaluates SCM-A2G-QFL using accuracy, validation performance, update norm, volatility, manifold dispersion, and SCM residual.The evaluation framework includes comparisons among FedAvg, QoS-weighted aggregation, circular A2G, MP-A2G, and SCM-A2G.
Key Contributions: SCM-A2G-QFL targets the coupled reliability–geometry problem that existing classical, manifold-aware, and QFL aggregation methods do not fully resolve.The paper contrasts its framework with FedAvg, FedProx, SCAFFOLD, circular aggregation, and midpoint methods designed for other settings.
Key Contributions: SCM-A2G-QFL treats the server movement from the current global model as the stabilization target, rather than only locating a geometry-aware client mean.This update-centric distinction addresses movements that may be geometrically valid yet too aggressive or unstable under non-IID data, quantum noise, and heterogeneous communication quality.
Key Contributions: The framework computes reliability-aware client weights, constructs a geometry-aware candidate direction, and refines the accepted global update using midpoint self-consistency.Clients return QNN parameters and QoS-related indicators at each communication round before server-side processing.
Key Contributions: The method assigns greater aggregation influence to clients with high channel fidelity, low latency, stable local behaviour, and sufficient data support.Unreliable communication or unstable updates are down-weighted before geometry-aware aggregation and midpoint refinement.

A. Geometry-Aware Candidate Update

The method forms a geometry-aware candidate movement using wrapped angular differences and QoS-weighted client directions, then refines that movement through midpoint self-consistency before updating the global model.

Candidate direction: Wrapped angular differences preserve torus geometry when constructing each client’s local direction.This avoids misleading Euclidean subtraction near the −π/π angular seam.
Candidate direction: The server combines local directions using normalized QoS trust weights and scales the initial candidate movement with geometry gain βt.The one-shot candidate is βtvt, where βt controls server movement size.
Candidate direction: A one-shot circular A2G update respects angular periodicity but may become unsupported after the server begins moving.This motivates midpoint and self-consistency refinement in noisy and heterogeneous QFL.
Midpoint refinement: SCM evaluates client directions at the midpoint induced by a candidate movement and forms a midpoint-supported QoS-weighted direction.The resulting map ψt(u) can differ from the direction computed at the current global point.
Global update: The next global model uses the accepted self-consistent movement rather than merely the first QoS-weighted direction.The update is applied from the current global model after midpoint-supported refinement.
Midpoint refinement: Under-relaxed fixed-point refinement adjusts the movement until it satisfies midpoint self-consistency or reaches the stopping conditions.The final iterate is accepted as u⋆t, and the SCM residual measures its remaining fixed-point error.

IV. CONVERGENCE ANALYSIS

The convergence analysis characterizes SCM-A2G-QFL as a geometry-aware descent step under non-convex federated objectives, with error controlled by several bounded sources.

Convergence analysis: SCM-A2G-QFL behaves as a geometry-aware descent step under a general non-convex federated objective.The analysis controls error from client heterogeneity, QoS-weight variance, angular wrapping, and SCM fixed-point residuals.
Evaluation diagnostics: The training procedure records global accuracy, validation loss, update norm, manifold dispersion, weight entropy, and SCM residual during communication rounds.These diagnostics accompany client selection, local training, QoS weighting, and server updates.

A. Assumptions

The analysis assumes smooth objectives, unbiased stochastic gradients, bounded client heterogeneity, bounded QoS weights, and bounded angular movement with SCM residual contribution.

A. Assumptions: The global and local objectives are assumed L-smooth in a tangent chart induced by wrapped angular differences.The QNN parameter vector is torus-valued, while analysis uses local tangent coordinates.
A. Assumptions: Client stochastic gradient estimators are assumed unbiased.This is stated as the unbiased local stochastic gradients assumption.
A. Assumptions: Gradient dissimilarity across clients is assumed bounded to model heterogeneous federated optimization.The assumption follows standard heterogeneous FL analyses.
A. Assumptions: QoS-aware weights are nonnegative, normalized, individually bounded, and have bounded deviation from data-size priors.These constraints limit the effect of reliability-based reweighting.
A. Assumptions: Wrapped local client directions and accepted SCM server movements are assumed bounded, with a bounded descent-scale SCM residual contribution.Angular wrapping, bounded geometry gain βt, and solver tolerance encourage the residual condition.

B. SCM Update Error Decomposition

SCM-A2G’s accepted movement is compared with the ideal centralized descent direction and decomposed into bounded errors from optimization, heterogeneity, QoS reweighting, midpoint displacement, and SCM residual effects.

B. SCM Update Error Decomposition: The ideal centralized descent direction at round t is −∇F(θt), whereas SCM-A2G applies an accepted server movement.The analysis evaluates the difference between these two directions.
B. SCM Update Error Decomposition: The training algorithm selects clients, broadcasts the global QNN, performs private local training, estimates QoS indicators, computes QoS weights, and applies the SCM server update.It also records accuracy, loss, update norm, manifold dispersion, weight entropy, and SCM residual diagnostics.
B. SCM Update Error Decomposition: The SCM fixed-point residual is a descent-scale contribution measuring remaining error in the midpoint solver.It quantifies how much this residual enters the one-step descent inequality.
B. SCM Update Error Decomposition: The update error includes stochastic local optimization error, client heterogeneity error, QoS reweighting bias, and midpoint displacement error.These terms are denoted ξl,t, ξg,t, ξq,t, and ξm,t, respectively.

C. Descent Lemma

The descent analysis bounds SCM-A2G progress under smoothness and related assumptions, accounting for stochastic, heterogeneity, QoS, geometry, and fixed-point errors.

C. Descent Lemma: Under Assumptions 1–5 and 0 < βt ≤ 1/L, Lemma IV.6 establishes one-step descent for the SCM-A2G update.The proof applies the L-smoothness inequality in the local tangent chart to the SCM update.
C. Descent Lemma: The descent bound accounts for stochastic local optimization, client heterogeneity, QoS-induced weighting, midpoint approximation, and fixed-point residual errors.These error sources are bounded using the stated assumptions and the SCM update decomposition.
C. Descent Lemma: Theorem IV.7 states non-convex convergence for SCM-A2G-QFL after T communication rounds when βt = β and 0 < β ≤ 1/L.F⋆ is a lower bound of the global objective, and C1, C2, C3 > 0 are constants independent of T.
C. Descent Lemma: If the SCM residual is uniformly bounded by RSCM,t ≤ εSCM, the convergence result includes a corresponding residual contribution.The residual bound is introduced as an additional condition on the SCM fixed-point error.

E. Interpretation

The analysis interprets SCM-A2G as a stability-controlled movement rule rather than only an aggregation target. Angular tests and theory connect midpoint self-consistency, residual accuracy, and the geometry–progress trade-off.

E. Interpretation: The average squared gradient norm decreases at an O(1/T) rate up to a neighbourhood set by training, heterogeneity, QoS, geometry, and SCM residual errors.The convergence neighbourhood includes the listed error sources rather than vanishing entirely with communication rounds.
E. Interpretation: The midpoint fixed-point residual directly affects the final convergence neighbourhood because it is specific to the self-consistent midpoint mechanism.Improving the midpoint solve therefore changes the residual contribution in the convergence guarantee.
E. Interpretation: Smaller geometry gain β reduces midpoint displacement, but excessively small β increases the relative residual contribution ε²SCM/β.Thus β controls a stability–progress trade-off rather than acting as a purely stabilizing parameter.
E. Interpretation: The framework combines QoS-aware weighting, torus-consistent candidate directions, and midpoint self-consistency to control the path from θt to θt+1.The accepted movement must remain consistent with the QoS-weighted direction evaluated at its own midpoint.
E. Interpretation: The experiments use medical Breast-Lesions-USG and financial BAF datasets, with QNN-compatible features and Qiskit-based simulator and hardware tooling.Fixed random seeds are used for dataset splitting, client partitioning, parameter initialization, and stochastic optimizer behaviour where supported.
E. Interpretation: Controlled angular tests distinguish direct geometry-aware targets from bounded next-server movements under wrap-around configurations.Euclidean averaging can produce seam-inconsistent targets, whereas MP-A2G and SCM-A2G move conservatively from the current global angle; SCM residuals rapidly approach zero.

D. Results

Across medical and financial datasets, SCM-A2G improves the accuracy–stability–movement-control trade-off, with lower volatility and smaller server movements while maintaining competitive or stronger predictive and validation performance.

1) Accuracy–stability behaviour: SCM-A2G remains competitive on Breast-Lesions-USG and achieves the strongest late-round accuracy with the lowest volatility on BAF.On Breast-Lesions-USG it achieves the lowest volatility; on BAF it achieves both the highest late-round mean accuracy and lowest volatility.
2) Performance and movement-control analysis: SCM-A2G achieves competitive trajectory-level AUC with much smaller maximum server update norms than FedAvg on Breast-Lesions-USG.FedAvg can reach high selected accuracy, but its larger update magnitudes indicate a less controlled aggregation path.
2) Performance and movement-control analysis: SCM-A2G achieves the highest AUC on BAF while keeping the maximum update norm much smaller than FedAvg.This combines stronger trajectory-level predictive performance with more controlled server movement.
3) Predictive and validation behaviour: SCM-A2G shows stronger validation behaviour across rounds, including the highest final-round validation accuracy on BAF.Breast-Lesions-USG test curves are noisy under stochastic QNN training and non-IID clinical partitions, while validation behaviour remains stronger over most rounds.
4) Aggregation-path diagnostics: FedAvg produces the largest and most volatile server update norms, whereas SCM-A2G produces the most controlled update trajectory in both datasets.SCM-A2G iteratively recomputes midpoint-supported directions and accepts movement only after self-consistency is reached.
D. Results: The overall advantage is an accuracy–stability–movement-control trade-off rather than accuracy improvement alone.The consistent behaviour across Breast-Lesions-USG and BAF supports a domain-general claim rather than tuning to one application.
D. Results: SCM-A2G outperforms FEDCOMPASS and FedMRUR in both validation accuracy and validation loss.This supports an additional stabilization benefit beyond geometry-aware aggregation alone.

APPENDIX

Hardware validation shows that aggregation geometry changes the quantum observables realized by QNN angles, while the IBM experiment validates executable SCM-A2G outputs rather than full federated training.

IBM Hardware Validation: Wrap-around aggregation produces different hardware observables: Euclidean averaging can yield positive expectations, whereas circular and SCM-based methods remain near the seam-consistent region.These methods produce negative observables close to simulator predictions in wrap-around cases.
Scope: The IBM experiment validates that SCM-A2G angles are executable on a real backend and preserve expected observable behavior, not that full federated QNN training ran on hardware.The reported experiments used one NVIDIA Tesla T4 GPU with CUDA 12.4 in a high-RAM runtime environment.
IBM Hardware Validation: In case B, Euclidean averaging gives θ = 0◦ and ZIBM = 0.9977, while circular, MP-A2G, and SCM-A2G produce negative expectations close to −1.
SCM Analysis: The SCM fixed-point relation decomposes the accepted server movement into a direction term and fixed-point residual error, with perturbation terms handled through inequalities and stated assumptions.The proof identifies stochastic optimization, client heterogeneity, QoS weighting, midpoint approximation, and residual error as components of the analysis.
SCM Analysis: The analysis concludes with Lemma IV.6 after substituting the decomposition and bounding the total perturbation term.

C. Proof of Theorem IV.7

The proof uses a constant geometry gain and bounded SCM residuals to derive the stated convergence rate, while the accompanying validation describes real-hardware observable evaluation.

Proof of Theorem IV.7: For constant geometry gain βt = β, summing the per-round bound over t = 0, . . . , T −1 yields the aggregate convergence inequality.
Proof of Theorem IV.7: Absorbing fixed numerical factors into C1, C2, and C3 produces Eq. (29), and residual control yields the simplified convergence rate in Eq. (30).
IBM Validation: Table VII evaluates controlled angular outputs by encoding θout in a single-qubit Ry(θ) circuit and measuring the Pauli-Z expectation on ibm_fez.Angles are reported in degrees, with Zsim as the ideal expectation and ZIBM as the hardware expectation; all transpiled circuits used zero CNOT gates.