A Stable Aggregation Method for Quantum Federated Learning

# A Stable Aggregation Method for Quantum Federated Learning
Source: [https://arxiv.org/html/2609.00356](https://arxiv.org/html/2609.00356)
Shanika Nanayakkara and Shiva Raj PokhrelAuthors are with the IoT & Software Engineering Research Lab, Deakin University, Geelong, VIC, Australia; email: s\.nanayakkara@deakin\.edu\.au, shiva\.pokhrel@deakin\.edu\.au

###### Abstract

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\.

## IIntroduction

Federated learning \(FL\)\[[19](https://arxiv.org/html/2609.00356#bib.bib25),[13](https://arxiv.org/html/2609.00356#bib.bib27),[9](https://arxiv.org/html/2609.00356#bib.bib2),[30](https://arxiv.org/html/2609.00356#bib.bib3)\]enables multiple clients to train a shared global model without exposing their private data\. Its practical performance, however, is strongly limited by the quality of server\-side aggregation\. Even in classical FL, with Federated Averaging \(FedAvg\[[19](https://arxiv.org/html/2609.00356#bib.bib25)\]\), the aggregation is affected by non\-IID data, unbalanced client samples, partial participation, and heterogeneous computing or communication resources\[[26](https://arxiv.org/html/2609.00356#bib.bib6),[25](https://arxiv.org/html/2609.00356#bib.bib7)\]\. These factors create biased local updates, client drift, and unstable convergence\. This has motivated robust aggregation and correction methods such as FedProx\[[16](https://arxiv.org/html/2609.00356#bib.bib9)\]and SCAFFOLD\[[13](https://arxiv.org/html/2609.00356#bib.bib27)\]\.

In quantum federated learning \(QFL\)\[[30](https://arxiv.org/html/2609.00356#bib.bib3),[5](https://arxiv.org/html/2609.00356#bib.bib18),[9](https://arxiv.org/html/2609.00356#bib.bib2)\], the aggregation problem illustrated in Fig[1](https://arxiv.org/html/2609.00356#S1.F1)becomes more challenging\. Observe in Fig[1](https://arxiv.org/html/2609.00356#S1.F1)that QFL extends federated optimization to distributed quantum and hybrid quantum–classical models, where clients train parameterized quantum circuits and the server aggregates quantum model parameters\[[5](https://arxiv.org/html/2609.00356#bib.bib18),[4](https://arxiv.org/html/2609.00356#bib.bib19),[29](https://arxiv.org/html/2609.00356#bib.bib20),[9](https://arxiv.org/html/2609.00356#bib.bib2)\]\. Unlike classical model parameters, many quantum neural network \(QNN\) parameters are rotation angles with periodic geometry\. Thus, direct Euclidean averaging can produce geometrically inconsistent updates, especially near angular wrap\-around boundaries\. Moreover, client updates in quantum\-enabled networks are affected not only by data heterogeneity but also by quantum\-channel fidelity, latency, decoherence, shot noise, and device instability\[[21](https://arxiv.org/html/2609.00356#bib.bib4)\]\. Therefore, QFL aggregation must be both reliability\-aware and geometry\-aware\. This motivates the need for aggregation rules that account for client quality while respecting the circular or torus\-valued structure of quantum model parameters\.

In such QFL settings, the quality of a client update may depend not only on data heterogeneity but also on quantum communication and device conditions, including teleportation fidelity\[[8](https://arxiv.org/html/2609.00356#bib.bib5)\], entanglement quality, latency, decoherence, and noisy intermediate\-scale quantum \(NISQ\) device instability\[[3](https://arxiv.org/html/2609.00356#bib.bib21),[14](https://arxiv.org/html/2609.00356#bib.bib22),[28](https://arxiv.org/html/2609.00356#bib.bib23),[24](https://arxiv.org/html/2609.00356#bib.bib24)\]\. Consequently, treating all client updates as equally reliable, or aggregating them only according to local data size, can be suboptimal\. A QFL aggregation rule should therefore account for both learning\-side heterogeneity and physical\-layer reliability when determining the contribution of each client to the global quantum model\.

![Refer to caption](https://arxiv.org/html/2609.00356v1/Basic_QFL_v.png)Figure 1:Abstract view of Quantum Federated LearningA second and more QNN\-specific challenge arises from the geometry of variational quantum circuit parameters\. Many trainable QNN parameters are rotation angles, and therefore satisfy the periodic equivalenceθ≡θ\+2​π\\theta\\equiv\\theta\+2\\pi\. Therefore, parameters close to\+π\+\\piand−π\-\\piare geometrically adjacent on the quantum parameter circle, even though their Euclidean coordinates appear far apart\. Naive Euclidean averaging can therefore move the aggregate toward an artificial value near zero, whereas circular aggregation preserves the correct angular geometry\[[18](https://arxiv.org/html/2609.00356#bib.bib45),[7](https://arxiv.org/html/2609.00356#bib.bib46)\]and enables adaptive aggregation\[[20](https://arxiv.org/html/2609.00356#bib.bib54)\]\.

### I\-AQFL Aggregation Problem and Ideas

Fig\.[2](https://arxiv.org/html/2609.00356#S1.F2)illustrates the angular aggregation problem using a single\-parameter slice of the QNN parameter space\. Although a full QNN model contains many rotation parameters, the one\-dimensional example captures the essential issue\. The ordinary Euclidean mean maps seam\-adjacent client angles to14\.35∘14\.35^\{\\circ\}, whereas the circular mean remains near176\.4∘176\.4^\{\\circ\}, which is geometrically consistent with the client cluster around the−180∘/\+180∘\-180^\{\\circ\}/\+180^\{\\circ\}seam\.

To this end, we introduce the following three big ideas\. We develop Adaptive Aggregation with two Gains \(A2G\)111A preliminary version of A2G is presented in the International Conference on Quantum Communications, Networking, and Computing \(QCNC 2026\)\[[20](https://arxiv.org/html/2609.00356#bib.bib54)\]\.as a novel framework that uses client reliability and update\-control gains to compute a stable global model update in quantum federated learning\.

∘\\circQoS\-weighted aggregation:QoS\-weighted aggregation gives more importance to clients with more reliable updates\. The client weight depends on data size, quantum fidelity, latency, and update stability\. Reliable clients contribute more to the global model, while noisy or delayed clients contribute less\.

∘\\circCircular A2G:Circular A2G is an adaptive aggregation method for quantum model parameters that are angles\. It uses circular geometry instead of ordinary Euclidean averaging\. This avoids wrong updates when angles are close to the−π/π\-\\pi/\\piwrap\-around boundary\.

∘\\circMP\-A2G:MP\-A2G means midpoint\-projected A2G\. It first computes a geometry\-aware aggregation direction and then applies a one\-shot midpoint\-based correction before updating the server model\. This aims to reduce overly aggressive movement toward an unstable aggregate\. We discuss details in Sec III later\.

Importantly, the A2G circular update, midpoint\-projected A2G update, and SCM\-A2G update do not collapse toward the misleading Euclidean aggregate\. Instead, starting from the current global pointθt=−170∘\\theta\_\{t\}=\-170^\{\\circ\}, they move conservatively toward the geometry\-consistent region of the angular parameter space\. This behaviour is essential: it shows that the proposed updates respect the circular topology of the quantum parameter manifold and avoid artificial averaging artifacts caused by treating angles as ordinary Euclidean scalars\.

It should be noted that our idea of angular discrepancy has a direct hardware\-level interpretation, as shown in Table[I](https://arxiv.org/html/2609.00356#S1.T1)\. In our IBM quantum hardware validation, each aggregated angle is encoded into a single\-qubitRy​\(θ\)R\_\{y\}\(\\theta\)circuit and evaluated using the Pauli\-ZZexpectation value\. The Euclidean mean yields a positive hardware measurement, indicating that it realizes a physically different quantum state from the geometry\-consistent solutions\. In contrast, the circular, midpoint\-projected, and SCM\-based updates produce negative IBM hardware measurements that closely match the corresponding simulator values\. This demonstrates that the choice of aggregation geometry is not merely a mathematical detail; it directly determines the quantum state implemented on real hardware\.

Therefore, geometry\-aware consistent aggregation is both theoretically necessary and experimentally meaningful\. By preserving the intrinsic angular structure of the parameter space, the proposed A2G and SCM\-A2G updates avoid spurious Euclidean averaging, maintain consistency with the intended quantum evolution, and produce hardware\-realized states that align with the true circular geometry of the model\.

### I\-BWhy Midpoint Self\-Consistency is Needed

![Refer to caption](https://arxiv.org/html/2609.00356v1/level1_full_circle_largefont.png)Figure 2:Motivating angular\-seam example for QNN parameter aggregation\. The current global parameter isθt=−170∘\\theta\_\{t\}=\-170^\{\\circ\}, while client angles lie near the−180∘/\+180∘\-180^\{\\circ\}/\+180^\{\\circ\}seam\. Euclidean averaging maps these seam\-adjacent angles to14\.35∘14\.35^\{\\circ\}, incorrectly placing the aggregate on the opposite side of the circle\. In contrast, circular aggregation, midpoint\-projected A2G, and SCM\-A2G remain in the geometry\-consistent angular region\.TABLE I:IBM hardware validation of geometry\-aware aggregation updates onibm\_fez\.θ\\thetaandΔ​θ\\Delta\\thetaare reported in degrees\.

Circular aggregation corrects the angular seam artefact, but it still computes the client\-supported direction mainly from the current global model\. In noisy and heterogeneous QFL, this first direction may be affected by non\-IID client updates, stochastic local QNN training, finite\-shot noise, and QoS variation\. A direction that appears suitable at the current point may become less reliable along the induced movement\.

To address this issue, as shown in the Figure[3](https://arxiv.org/html/2609.00356#S1.F3)midpoint\-projected aggregation introduces an intermediate check\. Before accepting the final server update, the server evaluates a midpoint along the candidate movement and recomputes the client\-supported direction from that midpoint\. This midpoint decision acts as a geometry\-aware stability check: it tests whether the proposed movement remains meaningful after the server has begun to move\.

The proposed self\-consistent midpoint aggregation further strengthens this idea\. Instead of applying only a one\-shot midpoint correction, SCM\-A2G accepts a server movement only when the movement is supported by its own midpoint\. Thus, the next global model is not obtained by direct Euclidean averaging, nor merely by a circular client mean\. It is obtained as a QoS\-weighted, torus\-aware, self\-consistent movement from the current global model\.

TABLE II:Main abbreviations used in this work\.![Refer to caption](https://arxiv.org/html/2609.00356v1/SCM_midpointProcess.png)Figure 3:SCM \(Self\-Consistent Midpoint\) Aggregation
### I\-CKey Contributions

Our main contributions are as follows\.

1. 1\.We identify a key aggregation challenge in QFL: client reliability varies, and QNN parameters lie on periodic angular spaces\. We show that Euclidean averaging can fail near angular seams and can lead to different quantum observables on IBM hardware\.
2. 2\.We propose SCM\-A2G\-QFL, a QoS\-aware and geometry\-aware aggregation method with midpoint self\-consistency\.

We show how FedAvg, QoS\-weighted aggregation, circular A2G, MP\-A2G, and SCM\-A2G fit into one aggregation framework\. We evaluate SCM\-A2G\-QFL using accuracy, validation performance, update norm, volatility, manifold dispersion, and SCM residual\.

## IIRelated Work

Existing aggregation methods address important parts of federated learning, but they do not fully resolve the coupled reliability–geometry problem that arises in QNN\-based QFL\. Classical FL methods such as FedAvg\[[19](https://arxiv.org/html/2609.00356#bib.bib25)\], FedProx\[[16](https://arxiv.org/html/2609.00356#bib.bib9)\], and SCAFFOLD\[[13](https://arxiv.org/html/2609.00356#bib.bib27)\]mainly assume that model parameters can be aggregated in Euclidean space\. FedAvg performs direct weighted model averaging, FedProx introduces a proximal local objective to reduce client drift, and SCAFFOLD uses control variates to correct client drift\. These methods are effective for many classical FL settings, but they do not explicitly account for periodic QNN parameters, whereθ≡θ\+2​π\\theta\\equiv\\theta\+2\\pi, and therefore may suffer from angular seam artefacts when applied directly to variational quantum circuits\.

Circular statistics and Riemannian averaging provide important foundations for non\-Euclidean aggregation\. Circular means avoid angular seam artefacts by averaging sine–cosine embeddings instead of raw angle values\[[18](https://arxiv.org/html/2609.00356#bib.bib45),[7](https://arxiv.org/html/2609.00356#bib.bib46)\]\. Fréchet/Karcher means generalize Euclidean averaging to manifold\-valued data by minimizing geodesic distances\[[17](https://arxiv.org/html/2609.00356#bib.bib44)\], while Riemannian FL extends federated optimization to manifold\-constrained models\[[15](https://arxiv.org/html/2609.00356#bib.bib13),[10](https://arxiv.org/html/2609.00356#bib.bib49)\]\. However, these approaches primarily answer the question of where the geometry\-aware mean or manifold optimizer lies\. They do not directly define a QoS\-aware server movement from the current global QNN model to the next global model, nor do they require that the accepted movement remain supported by its own midpoint\.

Midpoint\-based correction is also related to numerical methods for manifold\-constrained dynamics\. For example, midpoint projection has been used to improve the stability of stochastic differential equation integration on manifolds\[[12](https://arxiv.org/html/2609.00356#bib.bib16)\]\. However, such methods are not designed for federated aggregation, QoS\-weighted client trust, or torus\-valued QNN parameter updates\. SCM\-A2G\-QFL adapts the midpoint principle to the QFL server\-side aggregation setting by making the accepted global movement self\-consistent with its own midpoint\.

Recent QFL frameworks further motivate the need for quantum\-specific aggregation\. Federated quantum machine learning has been studied in hybrid classical–quantum settings\[[6](https://arxiv.org/html/2609.00356#bib.bib15),[4](https://arxiv.org/html/2609.00356#bib.bib19)\], while recent periodic QFL aggregation approaches such as FEDCOMPASS\[[27](https://arxiv.org/html/2609.00356#bib.bib52)\]use circular aggregation for quantum parameters together with client clustering\. This confirms that periodicity is important in QFL\. Nevertheless, circular aggregation mainly identifies a periodic client target; it does not by itself control how far the server should move from the current global model in one communication round\. Similarly, manifold\-regularized FL methods such as FedMRUR\[[2](https://arxiv.org/html/2609.00356#bib.bib51)\]address model inconsistency and update\-norm reduction under data heterogeneity, but they are not designed for QoS\-aware torus\-valued QNN parameter aggregation\. Ahmad*et al\.*studied FL under statistical heterogeneity on Riemannian manifolds\[[1](https://arxiv.org/html/2609.00356#bib.bib12)\], while FedSPDnet extends geometry\-aware FL to SPDNet models on symmetric positive definite manifolds\[[22](https://arxiv.org/html/2609.00356#bib.bib14)\]\. These works support the importance of non\-Euclidean FL, but they do not address QoS\-aware torus\-valued QNN aggregation or midpoint self\-consistent server movement\.

In contrast, SCM\-A2G\-QFL treats the server update itself as the object of stabilization and update\-centric\. Circular and Riemannian averaging methods primarily answer where the geometry\-aware client mean is located\. In contrast, SCM\-A2G\-QFL asks which movement from the current global model should be accepted under QoS\-weighted client evidence and midpoint self\-consistency\. This distinction is important in QNN\-based QFL because a direct aggregate may be geometrically valid as an average but still too aggressive or unstable as a server update under non\-IID data, quantum noise, and heterogeneous communication quality\. Table[III](https://arxiv.org/html/2609.00356#S2.T3)summarizes the key limitations of representative methods relative to the proposed SCM\-A2G\-QFL framework\. Table[III](https://arxiv.org/html/2609.00356#S2.T3)summarizes the position of the proposed method relative to representative classical FL, manifold\-aware, and QFL aggregation approaches\.

TABLE III:Component\-level comparison of existing aggregation, manifold\-aware, and quantum federated learning approaches with the proposed SCM\-A2G\-QFL method\.![Refer to caption](https://arxiv.org/html/2609.00356v1/Proposed_Method.png)Figure 4:Proposed SCM\-A2G\-QFL aggregation pipeline\.TABLE IV:Key notation\.
## IIIProposed SCM\-A2G\-QFL Method

The SCM\-A2G\-QFL framework is developed specifically for quantum federated learning\. We addresses two interconnected challenges: heterogeneous client reliability and the non\-Euclidean geometry of quantum parameters\. At each communication round, clients locally train QNN models and return model parameters together with QoS\-related indicators\. The server then computes reliability\-aware client weights, constructs a geometry\-aware candidate direction, and refines the accepted global update using midpoint self\-consistency\.

Let𝜽t∈𝕋d\\boldsymbol\{\\theta\}\_\{t\}\\in\\mathbb\{T\}^\{d\}denote the global QNN parameter vector at communication roundtt, where𝕋d\\mathbb\{T\}^\{d\}denotes thedd\-dimensional torus induced by periodic quantum rotation parameters\. Let𝜽i,t\\boldsymbol\{\\theta\}\_\{i,t\}be the local model returned by clientii, and letpi=\|Di\|/∑j\|Dj\|p\_\{i\}=\|D\_\{i\}\|/\\sum\_\{j\}\|D\_\{j\}\|denote the data\-size prior of clientii\. The goal is to compute the next global parameter vector𝜽t\+1\\boldsymbol\{\\theta\}\_\{t\+1\}as a controlled, QoS\-weighted, geometry\-aware movement from𝜽t\\boldsymbol\{\\theta\}\_\{t\}\.

The first component of SCM\-A2G\-QFL is the QoS trust gain\. In heterogeneous QFL, not all client updates should contribute equally to the server aggregation\. A client may have a large local dataset but poor quantum\-channel reliability, high communication latency, or unstable local model behaviour\. Therefore, before constructing the global update direction, the server assigns each client a QoS\-aware trust weight\.

For clientiiat communication roundtt, letFi,tF\_\{i,t\}denote the estimated teleportation fidelity or quantum\-channel reliability,τi,t\\tau\_\{i,t\}denote the communication latency, andVi,tV\_\{i,t\}denote an instability measure, such as the variance of the fidelity estimate or the local model variation\. We define the QoS score as

qi,t=Fi,tα\(τi,t\+ϵ\)γ​\(Vi,t\+ϵ\)δ,q\_\{i,t\}=\\frac\{F\_\{i,t\}^\{\\alpha\}\}\{\(\\tau\_\{i,t\}\+\\epsilon\)^\{\\gamma\}\(V\_\{i,t\}\+\\epsilon\)^\{\\delta\}\},\(1\)whereα\\alphacontrols the fidelity gain,γ\\gammacontrols the latency penalty,δ\\deltacontrols the instability penalty, andϵ\>0\\epsilon\>0avoids division by zero\.

Let

pi=\|Di\|∑j=1N\|Dj\|p\_\{i\}=\\frac\{\|D\_\{i\}\|\}\{\\sum\_\{j=1\}^\{N\}\|D\_\{j\}\|\}\(2\)denote the data\-size prior of clientii, where\|Di\|\|D\_\{i\}\|is the number of local training samples\. The final normalized aggregation weight is then

wi,t=pi​qi,t∑j=1Npj​qj,t\.w\_\{i,t\}=\\frac\{p\_\{i\}q\_\{i,t\}\}\{\\sum\_\{j=1\}^\{N\}p\_\{j\}q\_\{j,t\}\}\.\(3\)
Thus, clients with high channel fidelity, low latency, stable local behaviour, and sufficient data support receive larger aggregation influence\. Conversely, clients with unreliable communication or unstable updates are down\-weighted before the geometry\-aware and SCM midpoint\-refinement stages\.

### III\-AGeometry\-Aware Candidate Update

After computing the QoS\-aware weights, the server constructs a geometry\-aware candidate movement\. In Euclidean FL, the server can directly average client parameter vectors\. However, QNN parameters are often rotation angles and therefore satisfy the periodic equivalenceθ≡θ\+2​π\\theta\\equiv\\theta\+2\\pi\. Consequently, ordinary Euclidean subtraction can be misleading near the−π/π\-\\pi/\\piangular seam\. To preserve the torus geometry of the QNN parameter space, we define the local client direction using the wrapped angular difference:

𝐯i,t=wrap\[−π,π\)⁡\(𝜽i,t−𝜽t\),\\mathbf\{v\}\_\{i,t\}=\\operatorname\{wrap\}\_\{\[\-\\pi,\\pi\)\}\\left\(\\boldsymbol\{\\theta\}\_\{i,t\}\-\\boldsymbol\{\\theta\}\_\{t\}\\right\),\(4\)where𝜽t\\boldsymbol\{\\theta\}\_\{t\}is the current global QNN parameter vector and𝜽i,t\\boldsymbol\{\\theta\}\_\{i,t\}is the locally trained parameter vector returned by clientiiat roundtt\.

The QoS\-weighted global direction is then computed as

𝐯t=∑i=1Nwi,t​𝐯i,t,\\mathbf\{v\}\_\{t\}=\\sum\_\{i=1\}^\{N\}w\_\{i,t\}\\mathbf\{v\}\_\{i,t\},\(5\)wherewi,tw\_\{i,t\}is the normalized QoS trust weight from Eq\. \([3](https://arxiv.org/html/2609.00356#S3.E3)\)\. The initial geometry\-aware candidate movement is

𝐮t\(0\)=βt​𝐯t,\\mathbf\{u\}\_\{t\}^\{\(0\)\}=\\beta\_\{t\}\\mathbf\{v\}\_\{t\},\(6\)whereβt∈\(0,1\]\\beta\_\{t\}\\in\(0,1\]is the geometry gain controlling the size of the server movement\.

A one\-shot circular A2G update would therefore be

𝜽t\+1A2G=wrap\[−π,π\)⁡\(𝜽t\+𝐮t\(0\)\)=wrap\[−π,π\)⁡\(𝜽t\+βt​𝐯t\)\.\\boldsymbol\{\\theta\}\_\{t\+1\}^\{\\mathrm\{A2G\}\}=\\operatorname\{wrap\}\_\{\[\-\\pi,\\pi\)\}\\left\(\\boldsymbol\{\\theta\}\_\{t\}\+\\mathbf\{u\}\_\{t\}^\{\(0\)\}\\right\)=\\operatorname\{wrap\}\_\{\[\-\\pi,\\pi\)\}\\left\(\\boldsymbol\{\\theta\}\_\{t\}\+\\beta\_\{t\}\\mathbf\{v\}\_\{t\}\\right\)\.\(7\)This update respects angular periodicity and avoids Euclidean seam artefacts\. However, the direction𝐯t\\mathbf\{v\}\_\{t\}is computed only from the current global point𝜽t\\boldsymbol\{\\theta\}\_\{t\}\. In noisy and heterogeneous QFL, this initial direction may no longer be sufficiently supported once the server begins to move\. This motivates the midpoint and self\-consistency refinement introduced next\.

### III\-BSelf\-Consistent Midpoint Refinement

The midpoint refinement is introduced to avoid accepting a server movement that is supported only at the current global point but becomes unreliable along the movement path\. The one\-shot A2G movement𝐮t\(0\)=βt​𝐯t\\mathbf\{u\}\_\{t\}^\{\(0\)\}=\\beta\_\{t\}\\mathbf\{v\}\_\{t\}is used only as the initial candidate\. SCM then treats the server movement as a variable𝐮\\mathbf\{u\}and refines it until the movement is supported by its own midpoint\. Given any candidate movement𝐮\\mathbf\{u\}, SCM defines the induced midpoint as

𝐦t​\(𝐮\)=wrap\[−π,π\)⁡\(𝜽t\+12​𝐮\)\.\\mathbf\{m\}\_\{t\}\(\\mathbf\{u\}\)=\\operatorname\{wrap\}\_\{\[\-\\pi,\\pi\)\}\\left\(\\boldsymbol\{\\theta\}\_\{t\}\+\\frac\{1\}\{2\}\\mathbf\{u\}\\right\)\.\(8\)
From this midpoint, the server recomputes the wrapped client directions:

𝐯~i,t​\(𝐮\)=wrap\[−π,π\)⁡\(𝜽i,t−𝐦t​\(𝐮\)\),\\tilde\{\\mathbf\{v\}\}\_\{i,t\}\(\\mathbf\{u\}\)=\\operatorname\{wrap\}\_\{\[\-\\pi,\\pi\)\}\\left\(\\boldsymbol\{\\theta\}\_\{i,t\}\-\\mathbf\{m\}\_\{t\}\(\\mathbf\{u\}\)\\right\),\(9\)and forms the QoS\-weighted midpoint\-supported direction

𝝍t​\(𝐮\)=∑i=1Nwi,t​𝐯~i,t​\(𝐮\)\.\\boldsymbol\{\\psi\}\_\{t\}\(\\mathbf\{u\}\)=\\sum\_\{i=1\}^\{N\}w\_\{i,t\}\\tilde\{\\mathbf\{v\}\}\_\{i,t\}\(\\mathbf\{u\}\)\.\(10\)The map𝝍t​\(𝐮\)\\boldsymbol\{\\psi\}\_\{t\}\(\\mathbf\{u\}\)may differ from the initial direction𝐯t\\mathbf\{v\}\_\{t\}, because the client directions are evaluated from the midpoint induced by𝐮\\mathbf\{u\}, rather than directly from𝜽t\\boldsymbol\{\\theta\}\_\{t\}\.

Here,𝐮\\mathbf\{u\}denotes a generic candidate server movement\. The final SCM\-accepted movement is denoted by𝐮t⋆\\mathbf\{u\}\_\{t\}^\{\\star\}, which is the candidate movement satisfying the midpoint fixed\-point condition\.

𝐮t⋆=βt​𝝍t​\(𝐮t⋆\)\.\\mathbf\{u\}\_\{t\}^\{\\star\}=\\beta\_\{t\}\\boldsymbol\{\\psi\}\_\{t\}\(\\mathbf\{u\}\_\{t\}^\{\\star\}\)\.\(11\)Thus, the accepted server movement is not merely the first QoS\-weighted direction computed at𝜽t\\boldsymbol\{\\theta\}\_\{t\}\. Instead, it is a movement whose own midpoint continues to support the accepted movement\. The next global model is then updated as

𝜽t\+1=wrap\[−π,π\)⁡\(𝜽t\+𝐮t⋆\)\.\\boldsymbol\{\\theta\}\_\{t\+1\}=\\operatorname\{wrap\}\_\{\[\-\\pi,\\pi\)\}\\left\(\\boldsymbol\{\\theta\}\_\{t\}\+\\mathbf\{u\}\_\{t\}^\{\\star\}\\right\)\.\(12\)
In practice,𝐮t⋆\\mathbf\{u\}\_\{t\}^\{\\star\}is approximated by an under\-relaxed fixed\-point iteration\. Starting from the geometry\-aware candidate movement𝐮t\(0\)=βt​𝐯t\\mathbf\{u\}\_\{t\}^\{\(0\)\}=\\beta\_\{t\}\\mathbf\{v\}\_\{t\}, SCM iterates

𝐮t\(r\+1\)=\(1−ω\)​𝐮t\(r\)\+ω​βt​𝝍t​\(𝐮t\(r\)\),\\mathbf\{u\}\_\{t\}^\{\(r\+1\)\}=\(1\-\\omega\)\\mathbf\{u\}\_\{t\}^\{\(r\)\}\+\\omega\\beta\_\{t\}\\boldsymbol\{\\psi\}\_\{t\}\(\\mathbf\{u\}\_\{t\}^\{\(r\)\}\),\(13\)whereω∈\(0,1\]\\omega\\in\(0,1\]is the solver relaxation factor\. If the midpoint recomputed direction disagrees with the current candidate movement, the iteration adjusts the movement through𝝍t​\(𝐮t\(r\)\)\\boldsymbol\{\\psi\}\_\{t\}\(\\mathbf\{u\}\_\{t\}^\{\(r\)\}\)\. Hence, SCM changes the accepted movement according to the QoS\-weighted client evidence observed from the midpoint, rather than accepting the one\-shot direction\.

The iteration stops when

‖wrap\[−π,π\)⁡\(𝐮t\(r\+1\)−𝐮t\(r\)\)‖2<εSCM,\\left\\\|\\operatorname\{wrap\}\_\{\[\-\\pi,\\pi\)\}\\left\(\\mathbf\{u\}\_\{t\}^\{\(r\+1\)\}\-\\mathbf\{u\}\_\{t\}^\{\(r\)\}\\right\)\\right\\\|\_\{2\}<\\varepsilon\_\{\\mathrm\{SCM\}\},\(14\)or when a maximum number of SCM iterations is reached\. The final iterate is then used as the accepted movement𝐮t⋆\\mathbf\{u\}\_\{t\}^\{\\star\}\.

We record the SCM residual as

RSCM,t=‖wrap\[−π,π\)⁡\(𝐮t⋆−βt​𝝍t​\(𝐮t⋆\)\)‖2,R\_\{\\mathrm\{SCM\},t\}=\\left\\\|\\operatorname\{wrap\}\_\{\[\-\\pi,\\pi\)\}\\left\(\\mathbf\{u\}\_\{t\}^\{\\star\}\-\\beta\_\{t\}\\boldsymbol\{\\psi\}\_\{t\}\(\\mathbf\{u\}\_\{t\}^\{\\star\}\)\\right\)\\right\\\|\_\{2\},\(15\)which measures how closely the accepted movement satisfies the midpoint self\-consistency condition\.

Algorithm 1QoS Trust Weight Computation1:Selected clients

𝒮t\\mathcal\{S\}\_\{t\}; local dataset sizes

\{\|Di\|\}i∈𝒮t\\\{\|D\_\{i\}\|\\\}\_\{i\\in\\mathcal\{S\}\_\{t\}\}; QoS indicators

\{Fi,t,τi,t,Vi,t\}i∈𝒮t\\\{F\_\{i,t\},\\tau\_\{i,t\},V\_\{i,t\}\\\}\_\{i\\in\\mathcal\{S\}\_\{t\}\}; QoS gains

α,γ,δ\\alpha,\\gamma,\\delta; numerical constant

ϵ\>0\\epsilon\>0
2:Normalized QoS\-aware weights

\{wi,t\}i∈𝒮t\\\{w\_\{i,t\}\\\}\_\{i\\in\\mathcal\{S\}\_\{t\}\}
3:foreach selected client

i∈𝒮ti\\in\\mathcal\{S\}\_\{t\}do

4:Compute data\-size prior:

pi,t=\|Di\|∑j∈𝒮t\|Dj\|\.p\_\{i,t\}=\\frac\{\|D\_\{i\}\|\}\{\\sum\_\{j\\in\\mathcal\{S\}\_\{t\}\}\|D\_\{j\}\|\}\.
5:Compute QoS score:

qi,t=Fi,tα\(τi,t\+ϵ\)γ​\(Vi,t\+ϵ\)δ\.q\_\{i,t\}=\\frac\{F\_\{i,t\}^\{\\alpha\}\}\{\(\\tau\_\{i,t\}\+\\epsilon\)^\{\\gamma\}\(V\_\{i,t\}\+\\epsilon\)^\{\\delta\}\}\.
6:Compute unnormalized trust:

ai,t=pi,t​qi,t\.a\_\{i,t\}=p\_\{i,t\}q\_\{i,t\}\.
7:endfor

8:foreach selected client

i∈𝒮ti\\in\\mathcal\{S\}\_\{t\}do

9:Normalize aggregation weight:

wi,t=ai,t∑j∈𝒮taj,t\.w\_\{i,t\}=\\frac\{a\_\{i,t\}\}\{\\sum\_\{j\\in\\mathcal\{S\}\_\{t\}\}a\_\{j,t\}\}\.
10:endfor

11:return

\{wi,t\}i∈𝒮t\\\{w\_\{i,t\}\\\}\_\{i\\in\\mathcal\{S\}\_\{t\}\}

Algorithm 2SCM\-A2G Server Update1:Current global model

𝜽t\\boldsymbol\{\\theta\}\_\{t\}; selected client models

\{𝜽i,t\}i∈𝒮t\\\{\\boldsymbol\{\\theta\}\_\{i,t\}\\\}\_\{i\\in\\mathcal\{S\}\_\{t\}\}; QoS weights

\{wi,t\}i∈𝒮t\\\{w\_\{i,t\}\\\}\_\{i\\in\\mathcal\{S\}\_\{t\}\}; geometry gain

βt\\beta\_\{t\}; relaxation

ω\\omega; maximum SCM iterations

RmaxR\_\{\\max\}; tolerance

εSCM\\varepsilon\_\{\\mathrm\{SCM\}\}
2:Next global model

𝜽t\+1\\boldsymbol\{\\theta\}\_\{t\+1\}
3:foreach selected client

i∈𝒮ti\\in\\mathcal\{S\}\_\{t\}do

4:Compute local wrapped direction:

𝐯i,t=wrap\[−π,π\)⁡\(𝜽i,t−𝜽t\)\.\\mathbf\{v\}\_\{i,t\}=\\operatorname\{wrap\}\_\{\[\-\\pi,\\pi\)\}\\left\(\\boldsymbol\{\\theta\}\_\{i,t\}\-\\boldsymbol\{\\theta\}\_\{t\}\\right\)\.
5:endfor

6:Compute QoS\-weighted direction:

𝐯t=∑i∈𝒮twi,t​𝐯i,t\.\\mathbf\{v\}\_\{t\}=\\sum\_\{i\\in\\mathcal\{S\}\_\{t\}\}w\_\{i,t\}\\mathbf\{v\}\_\{i,t\}\.
7:Initialize SCM movement:

𝐮t\(0\)=βt​𝐯t\.\\mathbf\{u\}\_\{t\}^\{\(0\)\}=\\beta\_\{t\}\\mathbf\{v\}\_\{t\}\.
8:for

r=0,…,Rmax−1r=0,\\ldots,R\_\{\\max\}\-1do

9:Compute midpoint:

𝐦t\(r\)=wrap\[−π,π\)⁡\(𝜽t\+12​𝐮t\(r\)\)\.\\mathbf\{m\}\_\{t\}^\{\(r\)\}=\\operatorname\{wrap\}\_\{\[\-\\pi,\\pi\)\}\\left\(\\boldsymbol\{\\theta\}\_\{t\}\+\\frac\{1\}\{2\}\\mathbf\{u\}\_\{t\}^\{\(r\)\}\\right\)\.
10:foreach selected client

i∈𝒮ti\\in\\mathcal\{S\}\_\{t\}do

11:Recompute direction from midpoint:

𝐯~i,t\(r\)=wrap\[−π,π\)⁡\(𝜽i,t−𝐦t\(r\)\)\.\\tilde\{\\mathbf\{v\}\}\_\{i,t\}^\{\(r\)\}=\\operatorname\{wrap\}\_\{\[\-\\pi,\\pi\)\}\\left\(\\boldsymbol\{\\theta\}\_\{i,t\}\-\\mathbf\{m\}\_\{t\}^\{\(r\)\}\\right\)\.
12:endfor

13:Compute midpoint\-supported direction:

𝝍t\(r\)=∑i∈𝒮twi,t​𝐯~i,t\(r\)\.\\boldsymbol\{\\psi\}\_\{t\}^\{\(r\)\}=\\sum\_\{i\\in\\mathcal\{S\}\_\{t\}\}w\_\{i,t\}\\tilde\{\\mathbf\{v\}\}\_\{i,t\}^\{\(r\)\}\.
14:Compute fixed\-point candidate:

𝐮fp,t\(r\)=βt​𝝍t\(r\)\.\\mathbf\{u\}\_\{\\mathrm\{fp\},t\}^\{\(r\)\}=\\beta\_\{t\}\\boldsymbol\{\\psi\}\_\{t\}^\{\(r\)\}\.
15:Apply under\-relaxed refinement:

𝐮t\(r\+1\)=\(1−ω\)​𝐮t\(r\)\+ω​𝐮fp,t\(r\)\.\\mathbf\{u\}\_\{t\}^\{\(r\+1\)\}=\(1\-\\omega\)\\mathbf\{u\}\_\{t\}^\{\(r\)\}\+\\omega\\mathbf\{u\}\_\{\\mathrm\{fp\},t\}^\{\(r\)\}\.
16:Compute fixed\-point change:

Δt\(r\)=‖wrap\[−π,π\)⁡\(𝐮t\(r\+1\)−𝐮t\(r\)\)‖2\.\\Delta\_\{t\}^\{\(r\)\}=\\left\\\|\\operatorname\{wrap\}\_\{\[\-\\pi,\\pi\)\}\\left\(\\mathbf\{u\}\_\{t\}^\{\(r\+1\)\}\-\\mathbf\{u\}\_\{t\}^\{\(r\)\}\\right\)\\right\\\|\_\{2\}\.
17:if

Δt\(r\)<εSCM\\Delta\_\{t\}^\{\(r\)\}<\\varepsilon\_\{\\mathrm\{SCM\}\}then

18:break

19:endif

20:endfor

21:Set accepted movement:

𝐮t⋆=𝐮t\(r\+1\)\.\\mathbf\{u\}\_\{t\}^\{\\star\}=\\mathbf\{u\}\_\{t\}^\{\(r\+1\)\}\.
22:Update global model:

𝜽t\+1=wrap\[−π,π\)⁡\(𝜽t\+𝐮t⋆\)\.\\boldsymbol\{\\theta\}\_\{t\+1\}=\\operatorname\{wrap\}\_\{\[\-\\pi,\\pi\)\}\\left\(\\boldsymbol\{\\theta\}\_\{t\}\+\\mathbf\{u\}\_\{t\}^\{\\star\}\\right\)\.
23:return

𝜽t\+1\\boldsymbol\{\\theta\}\_\{t\+1\}

Algorithm 3SCM\-A2G\-QFL Training Procedure1:Initial global QNN parameters

𝜽0\\boldsymbol\{\\theta\}\_\{0\}; clients

\{1,…,N\}\\\{1,\\ldots,N\\\}; communication rounds

TT; local training routine

ℒ\\mathcal\{L\}; QoS gains

α,γ,δ\\alpha,\\gamma,\\delta; geometry gain

βt\\beta\_\{t\}; SCM parameters

Rmax,ω,εSCMR\_\{\\max\},\\omega,\\varepsilon\_\{\\mathrm\{SCM\}\}
2:Final global model

𝜽T\\boldsymbol\{\\theta\}\_\{T\}
3:for

t=0,…,T−1t=0,\\ldots,T\-1do

4:Select participating clients

𝒮t⊆\{1,…,N\}\\mathcal\{S\}\_\{t\}\\subseteq\\\{1,\\ldots,N\\\}\.

5:Server broadcasts

𝜽t\\boldsymbol\{\\theta\}\_\{t\}to clients in

𝒮t\\mathcal\{S\}\_\{t\}\.

6:foreach selected client

i∈𝒮ti\\in\\mathcal\{S\}\_\{t\}in paralleldo

7:Initialize local QNN with

𝜽t\\boldsymbol\{\\theta\}\_\{t\}\.

8:Train locally using routine

ℒ\\mathcal\{L\}on private data

DiD\_\{i\}\.

9:Return local model

𝜽i,t\\boldsymbol\{\\theta\}\_\{i,t\}\.

10:Estimate QoS indicators

Fi,tF\_\{i,t\},

τi,t\\tau\_\{i,t\}, and

Vi,tV\_\{i,t\}\.

11:endfor

12:Compute QoS\-aware weights

\{wi,t\}i∈𝒮t\\\{w\_\{i,t\}\\\}\_\{i\\in\\mathcal\{S\}\_\{t\}\}using Algorithm[1](https://arxiv.org/html/2609.00356#alg1)\.

13:Compute

𝜽t\+1\\boldsymbol\{\\theta\}\_\{t\+1\}using Algorithm[2](https://arxiv.org/html/2609.00356#alg2)\.

14:Record diagnostics: global accuracy, validation loss, update norm, manifold dispersion, weight entropy, and SCM residual\.

15:endfor

16:return

𝜽T\\boldsymbol\{\\theta\}\_\{T\}

## IVConvergence Analysis

We analyze the convergence behaviour of SCM\-A2G\-QFL under a general non\-convex federated objective\. Detailed proofs are provided in the Appendix\. The analysis shows that the proposed update behaves as a geometry\-aware descent step whose error is controlled by client heterogeneity, QoS\-weight variance, angular wrapping error, and the SCM fixed\-point residual\.

### IV\-AAssumptions

Let the global objective be

F​\(𝜽\)=∑i=1Npi​Fi​\(𝜽\),pi=\|Di\|∑j=1N\|Dj\|,F\(\\boldsymbol\{\\theta\}\)=\\sum\_\{i=1\}^\{N\}p\_\{i\}F\_\{i\}\(\\boldsymbol\{\\theta\}\),\\qquad p\_\{i\}=\\frac\{\|D\_\{i\}\|\}\{\\sum\_\{j=1\}^\{N\}\|D\_\{j\}\|\},\(16\)whereFiF\_\{i\}is the local objective of clientii, and𝜽∈𝕋d\\boldsymbol\{\\theta\}\\in\\mathbb\{T\}^\{d\}denotes the torus\-valued QNN parameter vector\. For analysis, we work in a local tangent chart induced by the wrapped angular differencewrap\[−π,π\)⁡\(⋅\)\\operatorname\{wrap\}\_\{\[\-\\pi,\\pi\)\}\(\\cdot\)\.

###### Assumption IV\.1\.

Smoothness: Each local objectiveFiF\_\{i\}and the global objectiveFFareLL\-smooth in the local tangent chart induced by the wrapped angular difference\[[16](https://arxiv.org/html/2609.00356#bib.bib9)\]\., i\.e\.,

‖∇Fi​\(𝒂\)−∇Fi​\(𝒃\)‖≤L​‖wrap⁡\(𝒂−𝒃\)‖,\\\|\\nabla F\_\{i\}\(\\boldsymbol\{a\}\)\-\\nabla F\_\{i\}\(\\boldsymbol\{b\}\)\\\|\\leq L\\\|\\operatorname\{wrap\}\(\\boldsymbol\{a\}\-\\boldsymbol\{b\}\)\\\|,\(17\)and similarly forFF\.

###### Assumption IV\.2\.

Unbiased local stochastic gradients: For clientii, the stochastic gradient estimator𝐠i,t\\mathbf\{g\}\_\{i,t\}satisfies

𝔼​\[𝐠i,t\]=∇Fi​\(𝜽t\),𝔼​‖𝐠i,t−∇Fi​\(𝜽t\)‖2≤σl2\.\\mathbb\{E\}\[\\mathbf\{g\}\_\{i,t\}\]=\\nabla F\_\{i\}\(\\boldsymbol\{\\theta\}\_\{t\}\),\\qquad\\mathbb\{E\}\\\|\\mathbf\{g\}\_\{i,t\}\-\\nabla F\_\{i\}\(\\boldsymbol\{\\theta\}\_\{t\}\)\\\|^\{2\}\\leq\\sigma\_\{l\}^\{2\}\.\(18\)

###### Assumption IV\.3\.

Bounded client heterogeneity: The gradient dissimilarity across clients is bounded, as commonly assumed in heterogeneous FL analyses\[[16](https://arxiv.org/html/2609.00356#bib.bib9),[13](https://arxiv.org/html/2609.00356#bib.bib27)\]\.

𝔼​‖∇Fi​\(𝜽\)−∇F​\(𝜽\)‖2≤σg2\.\\mathbb\{E\}\\\|\\nabla F\_\{i\}\(\\boldsymbol\{\\theta\}\)\-\\nabla F\(\\boldsymbol\{\\theta\}\)\\\|^\{2\}\\leq\\sigma\_\{g\}^\{2\}\.\(19\)

###### Assumption IV\.4\.

Bounded QoS weights: The QoS\-aware aggregation weights satisfywi,t≥0,∑i=1Nwi,t=1,wi,t≤wmax\.w\_\{i,t\}\\geq 0,\\qquad\\sum\_\{i=1\}^\{N\}w\_\{i,t\}=1,\\qquad w\_\{i,t\}\\leq w\_\{\\max\}\.Moreover, the deviation between QoS weights and data\-size priors is bounded:

∑i=1N\|wi,t−pi\|≤ρq\.\\sum\_\{i=1\}^\{N\}\|w\_\{i,t\}\-p\_\{i\}\|\\leq\\rho\_\{q\}\.\(20\)

###### Assumption IV\.5\(Bounded angular movement and SCM residual contribution\)\.

The wrapped local client directions and accepted SCM server movements are bounded:

‖𝐯i,t‖≤G,‖𝐮t⋆‖≤U\.\\\|\\mathbf\{v\}\_\{i,t\}\\\|\\leq G,\\qquad\\\|\\mathbf\{u\}\_\{t\}^\{\\star\}\\\|\\leq U\.\(21\)Moreover, the SCM fixed\-point residual contributes a bounded descent\-scale error:

𝔼​\[⟨∇F​\(𝜽t\),𝐞SCM,t⟩\]≤βt4​𝔼​\[‖∇F​\(𝜽t\)‖2\]\+C3​RSCM,t2\.\\mathbb\{E\}\\\!\\left\[\\left\\langle\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\),\\mathbf\{e\}\_\{\\mathrm\{SCM\},t\}\\right\\rangle\\right\]\\leq\\frac\{\\beta\_\{t\}\}\{4\}\\mathbb\{E\}\\\!\\left\[\\\|\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\)\\\|^\{2\}\\right\]\+C\_\{3\}R\_\{\\mathrm\{SCM\},t\}^\{2\}\.\(22\)Here,RSCM,tR\_\{\\mathrm\{SCM\},t\}denotes the SCM residual contribution measured at the descent scale\. This condition is encouraged by angular wrapping, bounded geometry gainβt\\beta\_\{t\}, and the stopping tolerance used in the SCM fixed\-point solver\.

### IV\-BSCM Update Error Decomposition

The ideal centralized descent direction at roundttis−∇F​\(𝜽t\)\-\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\)\. SCM\-A2G instead applies the accepted server movement

𝐮t⋆=βt​𝝍t​\(𝐮t⋆\)\+𝐞SCM,t,\\mathbf\{u\}\_\{t\}^\{\\star\}=\\beta\_\{t\}\\boldsymbol\{\\psi\}\_\{t\}\(\\mathbf\{u\}\_\{t\}^\{\\star\}\)\+\\mathbf\{e\}\_\{\\mathrm\{SCM\},t\},\(23\)where

𝐞SCM,t=𝐮t⋆−βt​𝝍t​\(𝐮t⋆\),‖𝐞SCM,t‖=RSCM,t\.\\mathbf\{e\}\_\{\\mathrm\{SCM\},t\}=\\mathbf\{u\}\_\{t\}^\{\\star\}\-\\beta\_\{t\}\\boldsymbol\{\\psi\}\_\{t\}\(\\mathbf\{u\}\_\{t\}^\{\\star\}\),\\qquad\\\|\\mathbf\{e\}\_\{\\mathrm\{SCM\},t\}\\\|=R\_\{\\mathrm\{SCM\},t\}\.\(24\)In the convergence analysis,RSCM,tR\_\{\\mathrm\{SCM\},t\}denotes the corresponding descent\-scale residual contribution defined through Assumption[IV\.5](https://arxiv.org/html/2609.00356#S4.Thmtheorem5)\. Thus,RSCM,tR\_\{\\mathrm\{SCM\},t\}measures how much the remaining SCM fixed\-point error contributes to the one\-step descent inequality\.ndition\.

The midpoint\-supported direction can be decomposed as

𝝍t​\(𝐮t⋆\)=−∇F​\(𝜽t\)\+𝝃l,t\+𝝃g,t\+𝝃q,t\+𝝃m,t,\\boldsymbol\{\\psi\}\_\{t\}\(\\mathbf\{u\}\_\{t\}^\{\\star\}\)=\-\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\)\+\\boldsymbol\{\\xi\}\_\{l,t\}\+\\boldsymbol\{\\xi\}\_\{g,t\}\+\\boldsymbol\{\\xi\}\_\{q,t\}\+\\boldsymbol\{\\xi\}\_\{m,t\},\(25\)where𝝃l,t\\boldsymbol\{\\xi\}\_\{l,t\}denotes stochastic local optimization error,𝝃g,t\\boldsymbol\{\\xi\}\_\{g,t\}denotes client heterogeneity error,𝝃q,t\\boldsymbol\{\\xi\}\_\{q,t\}denotes QoS reweighting bias, and𝝃m,t\\boldsymbol\{\\xi\}\_\{m,t\}denotes midpoint displacement error\.

Under Assumptions 1–5, these terms are bounded in expectation by

𝔼​‖𝝃l,t‖2\\displaystyle\\mathbb\{E\}\\\|\\boldsymbol\{\\xi\}\_\{l,t\}\\\|^\{2\}≤Cl​σl2,\\displaystyle\\leq C\_\{l\}\\sigma\_\{l\}^\{2\},𝔼​‖𝝃g,t‖2\\displaystyle\\mathbb\{E\}\\\|\\boldsymbol\{\\xi\}\_\{g,t\}\\\|^\{2\}≤Cg​σg2,\\displaystyle\\leq C\_\{g\}\\sigma\_\{g\}^\{2\},\(26\)𝔼​‖𝝃q,t‖2\\displaystyle\\mathbb\{E\}\\\|\\boldsymbol\{\\xi\}\_\{q,t\}\\\|^\{2\}≤Cq​ρq2​G2,\\displaystyle\\leq C\_\{q\}\\rho\_\{q\}^\{2\}G^\{2\},𝔼​‖𝝃m,t‖2\\displaystyle\\mathbb\{E\}\\\|\\boldsymbol\{\\xi\}\_\{m,t\}\\\|^\{2\}≤Cm​L2​‖𝐮t⋆‖2\.\\displaystyle\\leq C\_\{m\}L^\{2\}\\\|\\mathbf\{u\}\_\{t\}^\{\\star\}\\\|^\{2\}\.

### IV\-CDescent Lemma

###### Lemma IV\.6\(One\-step descent under SCM\-A2G\)\.

Under Assumptions 1–5, if the geometry gain satisfies0<βt≤1/L0<\\beta\_\{t\}\\leq 1/L, then the SCM\-A2G update

𝜽t\+1=wrap\[−π,π\)⁡\(𝜽t\+𝐮t⋆\)\\boldsymbol\{\\theta\}\_\{t\+1\}=\\operatorname\{wrap\}\_\{\[\-\\pi,\\pi\)\}\\\!\\left\(\\boldsymbol\{\\theta\}\_\{t\}\+\\mathbf\{u\}\_\{t\}^\{\\star\}\\right\)\(27\)satisfies

𝔼​\[F​\(𝜽t\+1\)\]≤\\displaystyle\\mathbb\{E\}\\\!\\left\[F\(\\boldsymbol\{\\theta\}\_\{t\+1\}\)\\right\]\\leq𝔼​\[F​\(𝜽t\)\]−βt2​𝔼​\[‖∇F​\(𝜽t\)‖2\]\\displaystyle\\mathbb\{E\}\\\!\\left\[F\(\\boldsymbol\{\\theta\}\_\{t\}\)\\right\]\-\\frac\{\\beta\_\{t\}\}\{2\}\\mathbb\{E\}\\\!\\left\[\\left\\\|\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\)\\right\\\|^\{2\}\\right\]\(28\)\+C1​βt​\(σl2\+σg2\+ρq2​G2\)\\displaystyle\+C\_\{1\}\\beta\_\{t\}\\left\(\\sigma\_\{l\}^\{2\}\+\\sigma\_\{g\}^\{2\}\+\\rho\_\{q\}^\{2\}G^\{2\}\\right\)\+C2​βt​L2​𝔼​\[‖𝐮t⋆‖2\]\+C3​RSCM,t2\.\\displaystyle\+C\_\{2\}\\beta\_\{t\}L^\{2\}\\mathbb\{E\}\\\!\\left\[\\left\\\|\\mathbf\{u\}\_\{t\}^\{\\star\}\\right\\\|^\{2\}\\right\]\+C\_\{3\}R\_\{\\mathrm\{SCM\},t\}^\{2\}\.

###### Proof sketch\.

The proof applies theLL\-smoothness inequality in the local tangent chart, substitutes the SCM update decomposition in Eq\. \([25](https://arxiv.org/html/2609.00356#S4.E25)\), and bounds the stochastic local optimization error, client heterogeneity error, QoS\-induced weighting error, midpoint approximation error, and fixed\-point residual error using Assumptions 1–5\. ∎

### IV\-DMain Convergence Result

###### Theorem IV\.7\(Non\-convex convergence of SCM\-A2G\-QFL\)\.

Suppose Assumptions 1–5 hold and choose a constant geometry gainβt=β\\beta\_\{t\}=\\betasuch that0<β≤1/L0<\\beta\\leq 1/L\. Then, afterTTcommunication rounds, SCM\-A2G\-QFL satisfies

1T​∑t=0T−1\\displaystyle\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}𝔼​\[‖∇F​\(𝜽t\)‖2\]≤2​\(F​\(𝜽0\)−F⋆\)β​T\\displaystyle\\mathbb\{E\}\\\!\\left\[\\\|\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\)\\\|^\{2\}\\right\]\\leq\\;\\frac\{2\(F\(\\boldsymbol\{\\theta\}\_\{0\}\)\-F^\{\\star\}\)\}\{\\beta T\}\(29\)\+C1​\(σl2\+σg2\+ρq2​G2\)\+C2​L2​1T​∑t=0T−1𝔼​\[‖𝐮t⋆‖2\]\\displaystyle\+C\_\{1\}\(\\sigma\_\{l\}^\{2\}\+\\sigma\_\{g\}^\{2\}\+\\rho\_\{q\}^\{2\}G^\{2\}\)\+C\_\{2\}L^\{2\}\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\\\!\\left\[\\\|\\mathbf\{u\}\_\{t\}^\{\\star\}\\\|^\{2\}\\right\]\+C3β​T​∑t=0T−1RSCM,t2\.\\displaystyle\+\\frac\{C\_\{3\}\}\{\\beta T\}\\sum\_\{t=0\}^\{T\-1\}R\_\{\\mathrm\{SCM\},t\}^\{2\}\.
HereF⋆F^\{\\star\}is a lower bound of the global objective, andC1,C2,C3\>0C\_\{1\},C\_\{2\},C\_\{3\}\>0are constants independent ofTT\.

If the SCM residual is uniformly bounded byRSCM,t≤εSCMR\_\{\\mathrm\{SCM\},t\}\\leq\\varepsilon\_\{\\mathrm\{SCM\}\}, then

1T​∑t=0T−1𝔼​\[‖∇F​\(𝜽t\)‖2\]≤\\displaystyle\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\\\!\\left\[\\\|\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\)\\\|^\{2\}\\right\]\\leq𝒪​\(1β​T\)\+𝒪​\(σl2\+σg2\+ρq2​G2\)\\displaystyle\\;\\mathcal\{O\}\\\!\\left\(\\frac\{1\}\{\\beta T\}\\right\)\+\\mathcal\{O\}\\\!\\left\(\\sigma\_\{l\}^\{2\}\+\\sigma\_\{g\}^\{2\}\+\\rho\_\{q\}^\{2\}G^\{2\}\\right\)\(30\)\+𝒪​\(L2​U2¯\)\+𝒪​\(εSCM2β\)\.\\displaystyle\+\\mathcal\{O\}\\\!\\left\(L^\{2\}\\overline\{U^\{2\}\}\\right\)\+\\mathcal\{O\}\\\!\\left\(\\frac\{\\varepsilon\_\{\\mathrm\{SCM\}\}^\{2\}\}\{\\beta\}\\right\)\.
whereU2¯=1T​∑t=0T−1𝔼​‖𝐮t⋆‖2\.\\overline\{U^\{2\}\}=\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\\\|\\mathbf\{u\}\_\{t\}^\{\\star\}\\\|^\{2\}\.

###### Proof\.

The proof follows by applying the smoothness descent lemma to the accepted SCM\-A2G server update, decomposing the update error into stochastic local optimization error, client heterogeneity error, QoS\-induced weighting error, geometry approximation error, and SCM fixed\-point residual error\. Summing the resulting one\-step descent inequality overt=0,…,T−1t=0,\\ldots,T\-1, using the lower boundedness ofFF, and rearranging terms yields \([29](https://arxiv.org/html/2609.00356#S4.E29)\)\. ∎

### IV\-EInterpretation

Theorem[IV\.7](https://arxiv.org/html/2609.00356#S4.Thmtheorem7)shows that SCM\-A2G\-QFL retains the standard non\-convex FL convergence structure\. The average squared gradient norm decreases at an𝒪​\(1/T\)\\mathcal\{O\}\(1/T\)rate up to a neighbourhood determined by stochastic local training, client heterogeneity, QoS\-induced reweighting, geometry\-induced movement error, and the SCM fixed\-point residual\. The residual term1β​T​∑t=0T−1RSCM,t2\\frac\{1\}\{\\beta T\}\\sum\_\{t=0\}^\{T\-1\}R\_\{\\mathrm\{SCM\},t\}^\{2\}is specific to the proposed self\-consistent midpoint mechanism\. It shows that the accuracy of the midpoint fixed\-point solve directly affects the final convergence neighbourhood\.

The result also explains the empirical stability observed in the experiments\. The geometry gainβ\\betacontrols how aggressively the server moves from the current global parameter𝜽t\\boldsymbol\{\\theta\}\_\{t\}toward the QoS\-weighted geometry\-aware direction\. Smaller accepted movements can reduce the midpoint displacement termL2​U2¯L^\{2\}\\overline\{U^\{2\}\}, but excessively smallβ\\betacan increase the relative effect of the residual contributionεSCM2/β\\varepsilon\_\{\\mathrm\{SCM\}\}^\{2\}/\\beta\. Thus,β\\betarepresents a stability–progress trade\-off rather than a purely stabilizing parameter\.

Overall, SCM\-A2G does not merely change the aggregation target\. It controls the path by which the global QNN model moves from𝜽t\\boldsymbol\{\\theta\}\_\{t\}to𝜽t\+1\\boldsymbol\{\\theta\}\_\{t\+1\}, requiring the accepted movement to remain consistent with the QoS\-weighted direction evaluated at its own midpoint\. This provides a theoretical explanation for the reduced late\-round volatility observed in the experiments\.

## VExperimental Evaluation

### V\-ADatasets and Domain Generalization

We use two binary classification datasets from substantially different domains\. The Breast\-Lesions\-USG represents a medical ultrasound diagnosis task, where patient age and ultrasound descriptors are used to classify lesions as benign or malignant\[[23](https://arxiv.org/html/2609.00356#bib.bib10)\]\. Invalid entries are excluded, categorical features are label\-encoded, age is converted to numeric form, and principal component analysis \(PCA\) apply after standardization to obtaind=2d=2ord=5d=5QNN\-compatible features\.

The Bank Account Fraud \(BAF\) dataset serves as a benchmark for financial fraud detection\. BAF was introduced at NeurIPS 2022 as a privacy\-preserving, large\-scale resource\. It provides a realistic tabular benchmark for detecting fraud in bank account openings\.\[[11](https://arxiv.org/html/2609.00356#bib.bib11)\]\. The suite comprises six synthetic datasets generated from an anonymized real\-world fraud detection environment and addresses practical challenges, including class imbalance, temporal dynamics, and controlled bias variants\. Each dataset includes 32 input features and up to one million records\. The binary target variable isfraud\_bool\. In the conducted experiments, missing values are removed, categorical variables are ordinal encoded, and numerical features are standardized\. PCA is applied to obtaind=4d=4QNN\-compatible features\. For both datasets, train, validation, and test splits are performed prior to encoding, scaling, and PCA to prevent data leakage\.

### V\-BExperimental Configuration

All experiments are implemented in Python using Qiskit, Qiskit Aer, and Qiskit Machine Learning\. Qiskit Aer is used for simulator\-based QNN execution\. The main supporting libraries include NumPy, pandas, scikit\-learn, and matplotlib\. Fixed random seeds are used for dataset splitting, client partitioning, QNN parameter initialization, and stochastic optimizer behaviour wherever supported\. The main experimental configuration is summarized in Table[V](https://arxiv.org/html/2609.00356#S5.T5)\. Dataset\-specific adjustments are made when required by sample size, circuit dimension, and computational budget\.

TABLE V:Main experimental configuration used in the domain\-generalized A2G and SCM\-A2G\-QFL experiments\.
### V\-CControlled Angular Stress\-Test Validation

TABLE VI:Controlled angular stress\-test cases\. Angles are in degrees\. Euclidean aggregation can move to seam\-inconsistent regions in wrap\-around cases, whereas MP\-A2G and SCM produce bounded geometry\-aware server movements\.RSCMR\_\{\\mathrm\{SCM\}\}denotes the final SCM fixed\-point residual\. Here,θEuc\\theta\_\{\\mathrm\{Euc\}\}andθCirc\\theta\_\{\\mathrm\{Circ\}\}denote direct aggregation targets, whereasθMP\\theta\_\{\\mathrm\{MP\}\}andθSCM\\theta\_\{\\mathrm\{SCM\}\}denote the next server states obtained after applying bounded geometry\-aware movement from the current server parameterθt\\theta\_\{t\}\.θEuc\\theta\_\{\\mathrm\{Euc\}\}denotes the Euclidean mean,θCirc\\theta\_\{\\mathrm\{Circ\}\}denotes the circular target,θMP\\theta\_\{\\mathrm\{MP\}\}denotes the midpoint\-projected update, andθSCM\\theta\_\{\\mathrm\{SCM\}\}denotes the proposed self\-consistent midpoint update\.RSCMR\_\{\\mathrm\{SCM\}\}is the SCM fixed\-point residual\.

![Refer to caption](https://arxiv.org/html/2609.00356v1/scm_residual_traces.png)Figure 5:SCM fixed\-point residual across controlled angular stress\-test cases\. The residual decreases rapidly across SCM iterations, showing that the accepted server movement becomes self\-consistent with its own midpoint\. The convergence remains stable from simple non\-wrap cases to more challenging wrap\-around configurations\. These resutls are generated in Real Qiskit IBM\. Appendix provide more results\.Prior to evaluating comprehensive federated QNN training, the angular component is first isolated\. This is achieved by examining aggregation behavior through controlled one\-dimensional stress tests\. These scenarios represent coordinate\-wise QNN rotation parameters on the torus and enable direct assessment of whether the server update remains geometrically consistent near angular seams i\.e\., the wrap\-around boundary where\+π\+\\piand−π\-\\pidenote adjacent directions\.\.

Table[VI](https://arxiv.org/html/2609.00356#S5.T6)reports four controlled angular stress tests\. Case A is a non\-wrap control case, where the client angles are clustered away from the−180∘/\+180∘\-180^\{\\circ\}/\+180^\{\\circ\}seam\. Euclidean and circular aggregation therefore produce nearly identical targets, showing that the geometry\-aware formulation does not distort benign angular configurations\.

Cases B and D test wrap\-around behaviour, where the client angles are geometrically concentrated near the seam\. In these cases, Euclidean averaging produces seam\-inconsistent targets, such as0∘0^\{\\circ\}and58\.07∘58\.07^\{\\circ\}, while circular aggregation recovers the correct seam\-consistent target\. In contrast, MP\-A2G and SCM\-A2G do not directly jump to this target; they compute bounded server movements from the current global angle\. Thus, values near−171∘\-171^\{\\circ\}in Cases B and D reflect controlled relaxation rather than aggregation error\.

This distinction is central: circular aggregation identifies the geometry\-aware client target, whereas SCM\-A2G determines the accepted next server state\. The SCM update is constrained by the geometry gain and midpoint self\-consistency, requiring the accepted movement to remain aligned with the QoS\-weighted direction at its own midpoint\. Fig\.[5](https://arxiv.org/html/2609.00356#S5.F5)confirms this fixed\-point behaviour, as the SCM residual rapidly decreases to near zero in all cases\. These tests therefore show that SCM\-A2G preserves benign\-case behaviour, avoids seam\-induced Euclidean failures, and stabilizes the server update through midpoint fixed\-point refinement\.

### V\-DResults

#### V\-D1Accuracy–stability behaviour

Fig\.[6](https://arxiv.org/html/2609.00356#S5.F6)compares the accuracy–stability behaviour of the methods on Breast\-Lesions\-USG and BAF\. Figures \(a\) and \(c\) show the last\-5\-round mean accuracy, while \(b\) and \(d\) show the corresponding late\-round volatility\. On Breast\-Lesions\-USG, SCM\-A2G remains competitive in late\-round accuracy and achieves the lowest volatility, indicating a more stable global trajectory near convergence\. Although FedAvg and MP\-A2G can attain comparable accuracy in some runs, their late\-round behaviour is less stable\. On BAF, SCM\-A2G provides the strongest result, achieving both the highest late\-round mean accuracy and the lowest volatility\. Overall, these results show that midpoint self\-consistency improves the accuracy–stability trade\-off, especially in the BAF setting\.

![Refer to caption](https://arxiv.org/html/2609.00356v1/box_last5_mean_accuracy.png)\(a\)Breast\-Lesions\-USG
![Refer to caption](https://arxiv.org/html/2609.00356v1/box_last5_volatility.png)\(b\)Breast\-Lesions\-USG
![Refer to caption](https://arxiv.org/html/2609.00356v1/baf_box_last5_mean_accuracy.png)\(c\)BAF
![Refer to caption](https://arxiv.org/html/2609.00356v1/baf_box_last5_volatility.png)\(d\)BAF

Figure 6:Accuracy–stability behaviour on Breast\-Lesions\-USG and BAF\. \(a\) and \(c\) compare primary performance across seeds, while \(b\) and \(d\) report last\-5\-round volatility\. SCM\-A2G achieves the most favourable accuracy–stability trade\-off: it remains competitive on Breast\-Lesions\-USG and obtains stronger performance with lower volatility on BAF\.
#### V\-D2Performance and movement\-control analysis

![Refer to caption](https://arxiv.org/html/2609.00356v1/box_auc_all.png)\(a\)Breast\-Lesions
![Refer to caption](https://arxiv.org/html/2609.00356v1/scatter_primary_perf_vs_max_update_norm.png)\(b\)Breast\-Lesions
![Refer to caption](https://arxiv.org/html/2609.00356v1/baf_box_auc_all.png)\(c\)Baf Data AUC
![Refer to caption](https://arxiv.org/html/2609.00356v1/baf_scatter_primary_perf_vs_max_update_norm.png)\(d\)Baf update norm\.

Figure 7:Performance and movement\-control analysis on Breast\-Lesions\-USG and BAF\. Subfigures \(a\) and \(c\) report AUC over all federated rounds, while Subfigures \(b\) and \(d\) relate selected test accuracy to the maximum server update norm\. SCM\-A2G achieves competitive or superior trajectory\-level performance while requiring substantially smaller server movements than FedAvg, supporting the implicit damping effect of self\-consistent midpoint aggregation\.Figure[7](https://arxiv.org/html/2609.00356#S5.F7)shows how predictive behavior relates to movement\. control\. On the Breast\-Lesions\-USG dataset, SCM\-A2G achieves competitive trajectory\-level AUC while keeping the maximum server update norm much lower than FedAvg\. FedAvg sometimes reaches high selected accuracy, but this is linked to aggressive update magnitudes, which suggests a less controlled aggregation path\. A similar pattern appears on the BAF dataset, where SCM\-A2G achieves the highest AUC while keeping the maximum update norm much smaller than FedAvg\. These results support the main idea behind SCM\-A2G: the method does not just optimize for endpoint accuracy, but also controls the accepted server movement by using midpoint self\-consistency\.

#### V\-D3Predictive and validation behaviour

![Refer to caption](https://arxiv.org/html/2609.00356v1/plot_accuracy_mean_ci.png)\(a\)Breast\-Lesion
![Refer to caption](https://arxiv.org/html/2609.00356v1/plot_validation_accuracy.png)\(b\)Breast\-Lesion Val Acc
![Refer to caption](https://arxiv.org/html/2609.00356v1/Baf_plot_accuracy_mean_ci.png)\(c\)BAF
![Refer to caption](https://arxiv.org/html/2609.00356v1/baf_plot_validation_accuracy.png)\(d\)BAF Val Acc

Figure 8:Predictive and validation behaviour on Breast\-Lesions\-USG and BAF\. Subfigures \(a\) and \(b\) correspond to Breast\-Lesions\-USG, while Subfigures \(c\) and \(d\) correspond to BAF\. SCM\-A2G maintains competitive global test accuracy and shows stronger validation behaviour, indicating improved generalization under heterogeneous federated settings\.![Refer to caption](https://arxiv.org/html/2609.00356v1/plot_diag_update_norm.png)\(a\)Server update norm\.
![Refer to caption](https://arxiv.org/html/2609.00356v1/plot_diag_midpoint_shift_norm.png)\(b\)Midpoint shift norm\.
![Refer to caption](https://arxiv.org/html/2609.00356v1/baf_plot_diag_update_norm.png)\(c\)Server update norm\.
![Refer to caption](https://arxiv.org/html/2609.00356v1/baf_plot_diag_midpoint_shift_norm.png)\(d\)Midpoint shift norm\.

Figure 9:Aggregation\-path diagnostics on the Breast\-Lesions\-USG dataset and BAF dataset\.Figure[8](https://arxiv.org/html/2609.00356#S5.F8)shows the test and validation accuracy for each round\. The test curves for Breast\-Lesions\-USG are quite noisy, which reflects the stochastic nature of quantum neural network \(QNN\) training and the presence of non\-independent and identically distributed \(non\-IID\) clinical partitions\. Nevertheless, the validation curve demonstrates that SCM\-A2G maintains stronger generalization behaviour over most rounds\. For the BAF dataset, the trend is more pronounced: SCM\-A2G improves steadily and achieves the highest validation accuracy by the final rounds\. Collectively, these results indicate that midpoint self\-consistency does not merely reduce update magnitudes; it also contributes to the development of global models with improved validation performance across domains\.

#### V\-D4Aggregation\-path diagnostics

Fig\.[9](https://arxiv.org/html/2609.00356#S5.F9)explains the mechanism behind the observed stability\. In both Breast\-Lesions\-USG and BAF, FedAvg produces the largest and most volatile server update norms, indicating aggressive movement of the global model\. A2G and MP\-A2G reduce this effect, but SCM\-A2G produces the most controlled update trajectory\. The midpoint\-shift plots further distinguish SCM\-A2G from MP\-A2G: while MP\-A2G applies a one\-shot midpoint correction, SCM\-A2G iteratively recomputes midpoint\-supported directions and accepts the movement only after self\-consistency is reached\. The lower midpoint\-shift norm therefore provides direct empirical evidence that SCM\-A2G stabilizes the accepted server movement\.

Overall, the results show that SCM\-A2G is not simply an accuracy\-improving heuristic\. Its main advantage is the accuracy–stability–movement\-control trade\-off\. FedAvg can occasionally achieve high accuracy, but it does so with large and unstable update movements\. A2G improves geometry awareness, and MP\-A2G introduces a one\-shot midpoint correction, but SCM\-A2G further stabilizes the server trajectory by requiring the accepted movement to be supported by its own midpoint\. This leads to lower late\-round volatility, smaller update norms, and stronger validation behaviour, especially on the BAF dataset\. The consistent behaviour across Breast\-Lesions\-USG and BAF supports the claim that SCM\-A2G is domain\-general rather than tuned to a single application\.

![Refer to caption](https://arxiv.org/html/2609.00356v1/comparative_validation_accuracy_mean_minmax.png)\(a\)
![Refer to caption](https://arxiv.org/html/2609.00356v1/comparative_validation_loss_mean_minmax.png)\(b\)

Figure 10:Comparative validation performance of SCM\-A2G against geometry\-aware baselines on the Breast\-Lesions\-USG dataset\. \(a\) reports validation accuracy across communication rounds, while \(b\) reports validation loss\.Furthermore, Fig\.[10](https://arxiv.org/html/2609.00356#S5.F10)compares SCM\-A2G with two geometry\-aware baselines, FEDCOMPASS\[[27](https://arxiv.org/html/2609.00356#bib.bib52)\]and FedMRUR\[[2](https://arxiv.org/html/2609.00356#bib.bib51)\]\. Overall, SCM\-A2G outperforms the geometry\-aware baselines in both validation accuracy and validation loss, supporting the claim that midpoint self\-consistency provides an additional stabilization benefit beyond geometry\-aware aggregation alone\.

## VIConclusion

This paper introduced SCM\-A2G\-QFL, a self\-consistent midpoint aggregation framework for QoS\- and geometry\-aware quantum federated learning\. The proposed method addresses two coupled challenges in QFL: heterogeneous client reliability and periodic QNN parameter geometry\. Unlike conventional Euclidean aggregation, SCM\-A2G\-QFL computes QoS\-aware client weights, constructs a torus\-consistent candidate direction, and accepts the next global movement only when it remains supported by its own midpoint\. Theoretical analysis showed that the resulting update is normalized, geometry\-preserving, locally stable under small geometry gains, and implicitly damped relative to direct A2G movement\. IBM hardware validation further demonstrated that aggregation geometry can induce physically different quantum observables, confirming the practical relevance of geometry\-aware QNN aggregation\. Overall, SCM\-A2G\-QFL provides a principled server\-side aggregation mechanism for stabilizing quantum federated learning under noisy, heterogeneous, and geometry\-sensitive conditions\.

## References

- \[1\]\(2023\)Federated learning under statistical heterogeneity on riemannian manifolds\.InPacific\-Asia Conference on Knowledge Discovery and Data Mining,pp\. 380–392\.Cited by:[TABLE III](https://arxiv.org/html/2609.00356#S2.T3.6.7.6.1.1.1),[§II](https://arxiv.org/html/2609.00356#S2.p4.1)\.
- \[2\]X\. An, L\. Shen, H\. Hu, and Y\. Luo\(2023\)Federated learning with manifold regularization and normalized update reaggregation\.InAdvances in Neural Information Processing Systems,Cited by:[TABLE III](https://arxiv.org/html/2609.00356#S2.T3.6.8.7.1.1.1),[§II](https://arxiv.org/html/2609.00356#S2.p4.1),[§V\-D4](https://arxiv.org/html/2609.00356#S5.SS4.SSS4.p3.1)\.
- \[3\]C\. H\. Bennett, G\. Brassard, C\. Crépeau, R\. Jozsa, A\. Peres, and W\. K\. Wootters\(1993\)Teleporting an unknown quantum state via dual classical and einstein\-podolsky\-rosen channels\.Physical Review Letters70\(13\),pp\. 1895–1899\.External Links:[Document](https://dx.doi.org/10.1103/PhysRevLett.70.1895)Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p3.1)\.
- \[4\]M\. Chehimi, S\. Y\. Chen, W\. Saad, D\. Towsley, and M\. Debbah\(2023\)Foundations of quantum federated learning over classical and quantum networks\.arXiv preprint arXiv:2310\.14516\.Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p2.1),[§II](https://arxiv.org/html/2609.00356#S2.p4.1)\.
- \[5\]M\. Chehimi and W\. Saad\(2022\)Quantum federated learning with quantum data\.InIEEE International Conference on Acoustics, Speech and Signal Processing,pp\. 8617–8621\.Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p2.1)\.
- \[6\]S\. Y\. Chen and S\. Yoo\(2021\)Federated quantum machine learning\.Entropy23\(4\),pp\. 460\.Cited by:[§II](https://arxiv.org/html/2609.00356#S2.p4.1)\.
- \[7\]N\. I\. Fisher\(1993\)Statistical analysis of circular data\.Cambridge University Press,Cambridge, U\.K\.\.Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p4.3),[TABLE III](https://arxiv.org/html/2609.00356#S2.T3.6.5.4.1.1.1),[§II](https://arxiv.org/html/2609.00356#S2.p2.1)\.
- \[8\]D\. Gurung and S\. R\. Pokhrel\(2026\)Towards quantum teleportation based barrage relay networks\.In2026 40th International Conference on Information Networking \(ICOIN\),Vol\.,pp\. 587–590\.External Links:[Document](https://dx.doi.org/10.1109/ICOIN68469.2026.11480649)Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p3.1)\.
- \[9\]B\. Hazarikaet al\.\(2025\)Quantum\-enhanced federated learning for metaverse\-empowered vehicular networks\.IEEE Transactions on Communications73\(6\),pp\. 4168–4183\.External Links:[Document](https://dx.doi.org/10.1109/TCOMM.2024.3502667)Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p1.1),[§I](https://arxiv.org/html/2609.00356#S1.p2.1)\.
- \[10\]Z\. Huang, W\. Huang, P\. Jawanpuria, and B\. Mishra\(2026\)Riemannian federated learning via averaging gradient streams\.InProceedings of the International Conference on Learning Representations,Cited by:[TABLE III](https://arxiv.org/html/2609.00356#S2.T3.6.7.6.1.1.1),[§II](https://arxiv.org/html/2609.00356#S2.p2.1)\.
- \[11\]S\. Jesus, J\. Pombal, D\. Alves, A\. Cruz, P\. Saleiro, R\. Ribeiro, J\. Gama, and P\. Bizarro\(2022\)Turning the tables: biased, imbalanced, dynamic tabular datasets for ml evaluation\.Advances in Neural Information Processing Systems35,pp\. 33563–33575\.Cited by:[§V\-A](https://arxiv.org/html/2609.00356#S5.SS1.p2.1)\.
- \[12\]R\. R\. Joseph, J\. van Rhijn, and P\. D\. Drummond\(2023\)Midpoint projection algorithm for stochastic differential equations on manifolds\.Physical Review E107\(5\),pp\. 055307\.Cited by:[§II](https://arxiv.org/html/2609.00356#S2.p3.1)\.
- \[13\]S\. P\. Karimireddy, S\. Kale, M\. Mohri, S\. Reddi, S\. Stich, and A\. T\. Suresh\(2020\)SCAFFOLD: stochastic controlled averaging for federated learning\.InProceedings of the International Conference on Machine Learning,pp\. 5132–5143\.Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p1.1),[TABLE III](https://arxiv.org/html/2609.00356#S2.T3.6.4.3.1.1.1),[§II](https://arxiv.org/html/2609.00356#S2.p1.1),[Assumption IV\.3](https://arxiv.org/html/2609.00356#S4.Thmtheorem3.p1.1)\.
- \[14\]H\. J\. Kimble\(2008\)The quantum internet\.Nature453,pp\. 1023–1030\.External Links:[Document](https://dx.doi.org/10.1038/nature07127)Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p3.1)\.
- \[15\]J\. Li and S\. Ma\(2022\)Federated learning on riemannian manifolds\.arXiv preprint arXiv:2206\.05668\.Cited by:[§II](https://arxiv.org/html/2609.00356#S2.p2.1)\.
- \[16\]T\. Li, A\. K\. Sahu, A\. Talwalkar, and V\. Smith\(2020\)Federated optimization in heterogeneous networks\.InProceedings of Machine Learning and Systems,Vol\.2,pp\. 429–450\.Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p1.1),[TABLE III](https://arxiv.org/html/2609.00356#S2.T3.6.3.2.1.1.1),[§II](https://arxiv.org/html/2609.00356#S2.p1.1),[Assumption IV\.1](https://arxiv.org/html/2609.00356#S4.Thmtheorem1.p1.3),[Assumption IV\.3](https://arxiv.org/html/2609.00356#S4.Thmtheorem3.p1.1)\.
- \[17\]C\. Mancinelli and E\. Puppo\(2023\)Computing the riemannian center of mass on meshes\.Computer Aided Geometric Design103,pp\. 102203\.External Links:[Document](https://dx.doi.org/10.1016/j.cagd.2023.102203)Cited by:[TABLE III](https://arxiv.org/html/2609.00356#S2.T3.6.6.5.1.1.1),[§II](https://arxiv.org/html/2609.00356#S2.p2.1)\.
- \[18\]K\. V\. Mardia and P\. E\. Jupp\(2000\)Directional statistics\.Wiley,Chichester, U\.K\.\.Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p4.3),[TABLE III](https://arxiv.org/html/2609.00356#S2.T3.6.5.4.1.1.1),[§II](https://arxiv.org/html/2609.00356#S2.p2.1)\.
- \[19\]H\. B\. McMahan, E\. Moore, D\. Ramage, S\. Hampson, and B\. Agüera y Arcas\(2017\)Communication\-efficient learning of deep networks from decentralized data\.InProceedings of the International Conference on Artificial Intelligence and Statistics,pp\. 1273–1282\.Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p1.1),[TABLE III](https://arxiv.org/html/2609.00356#S2.T3.6.2.1.1.1.1),[§II](https://arxiv.org/html/2609.00356#S2.p1.1)\.
- \[20\]S\. I\. Nanayakkara and S\. R\. Pokhrel\(2026\)A2G\-qfl: adaptive aggregation with two gains in quantum federated learning\.In2026 International Conference on Quantum Communications, Networking, and Computing \(QCNC\),Vol\.,pp\. 808–812\.External Links:[Document](https://dx.doi.org/10.1109/QCNC69040.2026.00132)Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p4.3),[footnote 1](https://arxiv.org/html/2609.00356#footnote1)\.
- \[21\]A\. Paul and K\. Singh\(2026\)Large ai model\-driven quantum\-enhanced transformer\-vqc federated drl for privacy preservation in vehicular networks\.IEEE Journal on Selected Areas in Communications44\(\),pp\. 3021–3038\.External Links:[Document](https://dx.doi.org/10.1109/JSAC.2025.3647821)Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p2.1)\.
- \[22\]T\. Pautrel, F\. Bouchard, A\. Mian, and G\. Ginolhac\(2026\)FedSPDnet: geometry\-aware federated deep learning with spdnet\.arXiv preprint arXiv:2604\.22494\.Cited by:[TABLE III](https://arxiv.org/html/2609.00356#S2.T3.6.10.9.1.1.1),[§II](https://arxiv.org/html/2609.00356#S2.p4.1)\.
- \[23\]A\. Pawłowska, A\. Ćwierz\-Pieńkowska, A\. Domalik, D\. Jaguś, P\. Kasprzak, R\. Matkowski, Ł\. Fura, A\. Nowicki, and N\. Żołek\(2024\)Curated benchmark dataset for ultrasound based breast lesion analysis\.Scientific Data11\(1\),pp\. 148\.Cited by:[§V\-A](https://arxiv.org/html/2609.00356#S5.SS1.p1.2)\.
- \[24\]J\. Preskill\(2018\)Quantum computing in the NISQ era and beyond\.Quantum2,pp\. 79\.External Links:[Document](https://dx.doi.org/10.22331/q-2018-08-06-79)Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p3.1)\.
- \[25\]L\. Qiao, Z\. Gao, M\. Boloursaz Mashhadi, and D\. Gündüz\(2024\)Massive digital over\-the\-air computation for communication\-efficient federated edge learning\.IEEE Journal on Selected Areas in Communications42\(11\),pp\. 3078–3094\.External Links:[Document](https://dx.doi.org/10.1109/JSAC.2024.3431572)Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p1.1)\.
- \[26\]C\. Ren, Y\. Tang, Y\. Gao, X\. Sun, K\. Fu, M\. Skoglund, Z\. Y\. Dong, H\. Yu, A\. Li, and M\. Xiao\(2025\)QFEVAL: quantum federated ensembled variational adaptive learning for dynamic security assessment in cyber\-physical systems\.IEEE Journal on Selected Areas in Communications43\(9\),pp\. 3200–3213\.External Links:[Document](https://dx.doi.org/10.1109/JSAC.2025.3574588)Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p1.1)\.
- \[27\]Y\. Wang, X\. He, Z\. Cai, R\. Zhang, R\. Ma, Y\. Liu, and R\. Buyya\(2026\)FedCompass: federated clustered and periodic aggregation framework for hybrid classical\-quantum models\.External Links:2602\.03052Cited by:[TABLE III](https://arxiv.org/html/2609.00356#S2.T3.6.9.8.1.1.1),[§II](https://arxiv.org/html/2609.00356#S2.p4.1),[§V\-D4](https://arxiv.org/html/2609.00356#S5.SS4.SSS4.p3.1)\.
- \[28\]S\. Wehner, D\. Elkouss, and R\. Hanson\(2018\)Quantum internet: a vision for the road ahead\.Science362\(6412\),pp\. eaam9288\.External Links:[Document](https://dx.doi.org/10.1126/science.aam9288)Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p3.1)\.
- \[29\]Q\. Xia and Q\. Li\(2021\)QuantumFed: a federated learning framework for collaborative quantum training\.arXiv preprint arXiv:2106\.09109\.Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p2.1)\.
- \[30\]Q\. Xu, R\. Li, Y\. Qi, Z\. Su, and D\. Fang\(2025\)Trust\-enhanced game incentive for secure quantum federated learning in uav\-assisted wireless networks\.IEEE Journal on Selected Areas in Communications43\(8\),pp\. 2841–2856\.External Links:[Document](https://dx.doi.org/10.1109/JSAC.2025.3568054)Cited by:[§I](https://arxiv.org/html/2609.00356#S1.p1.1),[§I](https://arxiv.org/html/2609.00356#S1.p2.1)\.

### \-AIBM Hardware Validation

TABLE VII:IBM Quantum cloud real validation of controlled angular aggregation outputs onibm\_fez\. The output angleθout\\theta\_\{\\mathrm\{out\}\}is encoded as a single\-qubitRy​\(θ\)R\_\{y\}\(\\theta\)circuit and evaluated using the Pauli\-ZZexpectation\.Δ​θt\\Delta\\theta\_\{t\}denotes the accepted angular movement from the current server parameterθt=−170∘\\theta\_\{t\}=\-170^\{\\circ\}\.All angles are reported in degrees\.ZsimZ\_\{\\mathrm\{sim\}\}denotes the ideal simulator expectation andZIBMZ\_\{\\mathrm\{IBM\}\}denotes the IBM hardware expectation\.\|Δ​Z\|=\|ZIBM−Zsim\|\|\\Delta Z\|=\|Z\_\{\\mathrm\{IBM\}\}\-Z\_\{\\mathrm\{sim\}\}\|\. The SCM residual is not shown here because it is an algorithmic fixed\-point quantity rather than a hardware\-measured observable\. All transpiled circuits used zero CNOT gates on the selected backend layout\.

The hardware results show that Euclidean and circular/SCM\-based aggregation can produce substantially different quantum observables\. For wrap\-around cases, Euclidean aggregation may yield angles whose ideal observables are positive, whereas circular, MP\-A2G, and SCM\-A2G remain near the seam\-consistent region and produce negative observables close to the simulator prediction\. This confirms that the aggregation geometry affects the quantum STATE realised on hardware\.

Importantly, the IBM experiment is not intended to claim that full federated QNN training was executed on hardware\. Rather, it provides hardware\-backed validation that the angles generated by SCM\-A2G are executable on a real backend and preserve the expected quantum observable behaviour\. All experiments utilized a single NVIDIA Tesla T4 graphics processing unit \(GPU\) and CUDA version 12\.4 in a high\-RAM runtime environment\.

Table[VII](https://arxiv.org/html/2609.00356#A0.T7)validates that aggregation\-induced angles produce the expected observable behavior on real IBM hardware\. In the wrap\-around case B, Euclidean averaging givesθ=0∘\\theta=0^\{\\circ\}, producing a positive hardware expectationZIBM=0\.9977Z\_\{\\mathrm\{IBM\}\}=0\.9977\. In contrast, circular, MP\-A2G, and SCM\-A2G remain near the seam\-consistent region and produce negative expectations close to−1\-1\. Similarly, in case D, the Euclidean aggregate givesZIBM=0\.4963Z\_\{\\mathrm\{IBM\}\}=0\.4963, whereas circular and SCM\-based updates produce negative values close to simulator predictions\. These results show that aggregation geometry has a direct physical effect on the quantum observable\.

## Appendix AProof of Theorem[IV\.7](https://arxiv.org/html/2609.00356#S4.Thmtheorem7)

### A\-AProof of Lemma[IV\.6](https://arxiv.org/html/2609.00356#S4.Thmtheorem6)

Let

𝚵t=𝝃l,t\+𝝃g,t\+𝝃q,t\+𝝃m,t\\boldsymbol\{\\Xi\}\_\{t\}=\\boldsymbol\{\\xi\}\_\{l,t\}\+\\boldsymbol\{\\xi\}\_\{g,t\}\+\\boldsymbol\{\\xi\}\_\{q,t\}\+\\boldsymbol\{\\xi\}\_\{m,t\}denote the total perturbation induced by stochastic local optimization, client heterogeneity, QoS\-dependent weighting, and midpoint approximation\.

In the local tangent chart around𝜽t\\boldsymbol\{\\theta\}\_\{t\}, let𝐮t⋆\\mathbf\{u\}\_\{t\}^\{\\star\}denote the accepted SCM\-A2G server movement\. By theLL\-smoothness ofFF, we have

F​\(𝜽t\+1\)≤F​\(𝜽t\)\+⟨∇F​\(𝜽t\),𝐮t⋆⟩\+L2​‖𝐮t⋆‖2\.F\(\\boldsymbol\{\\theta\}\_\{t\+1\}\)\\leq F\(\\boldsymbol\{\\theta\}\_\{t\}\)\+\\left\\langle\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\),\\mathbf\{u\}\_\{t\}^\{\\star\}\\right\\rangle\+\\frac\{L\}\{2\}\\left\\\|\\mathbf\{u\}\_\{t\}^\{\\star\}\\right\\\|^\{2\}\.\(31\)
The SCM fixed\-point relation can be written as

𝐮t⋆=βt​𝝍t​\(𝐮t⋆\)\+𝐞SCM,t,\\mathbf\{u\}\_\{t\}^\{\\star\}=\\beta\_\{t\}\\boldsymbol\{\\psi\}\_\{t\}\(\\mathbf\{u\}\_\{t\}^\{\\star\}\)\+\\mathbf\{e\}\_\{\\mathrm\{SCM\},t\},\(32\)where𝐞SCM,t\\mathbf\{e\}\_\{\\mathrm\{SCM\},t\}denotes the fixed\-point residual error\. The direction term admits the decomposition

𝝍t​\(𝐮t⋆\)=−∇F​\(𝜽t\)\+𝚵t\.\\boldsymbol\{\\psi\}\_\{t\}\(\\mathbf\{u\}\_\{t\}^\{\\star\}\)=\-\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\)\+\\boldsymbol\{\\Xi\}\_\{t\}\.\(33\)
Substituting \([33](https://arxiv.org/html/2609.00356#A1.E33)\) into \([32](https://arxiv.org/html/2609.00356#A1.E32)\) gives

𝐮t⋆=−βt​∇F​\(𝜽t\)\+βt​𝚵t\+𝐞SCM,t\.\\mathbf\{u\}\_\{t\}^\{\\star\}=\-\\beta\_\{t\}\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\)\+\\beta\_\{t\}\\boldsymbol\{\\Xi\}\_\{t\}\+\\mathbf\{e\}\_\{\\mathrm\{SCM\},t\}\.\(34\)
Therefore,

⟨∇F​\(𝜽t\),𝐮t⋆⟩=\\displaystyle\\left\\langle\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\),\\mathbf\{u\}\_\{t\}^\{\\star\}\\right\\rangle=−βt​‖∇F​\(𝜽t\)‖2\\displaystyle\-\\beta\_\{t\}\\left\\\|\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\)\\right\\\|^\{2\}\(35\)\+βt​⟨∇F​\(𝜽t\),𝚵t⟩\\displaystyle\+\\beta\_\{t\}\\left\\langle\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\),\\boldsymbol\{\\Xi\}\_\{t\}\\right\\rangle\+⟨∇F​\(𝜽t\),𝐞SCM,t⟩\.\\displaystyle\+\\left\\langle\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\),\\mathbf\{e\}\_\{\\mathrm\{SCM\},t\}\\right\\rangle\.
Using Young’s inequality and Assumptions 1–5, the total perturbation term satisfies

𝔼​\[βt​⟨∇F​\(𝜽t\),𝚵t⟩\]≤\\displaystyle\\mathbb\{E\}\\\!\\left\[\\beta\_\{t\}\\left\\langle\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\),\\boldsymbol\{\\Xi\}\_\{t\}\\right\\rangle\\right\]\\leqβt4​𝔼​\[‖∇F​\(𝜽t\)‖2\]\\displaystyle\\frac\{\\beta\_\{t\}\}\{4\}\\mathbb\{E\}\\\!\\left\[\\left\\\|\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\)\\right\\\|^\{2\}\\right\]\(36\)\+C1​βt​\(σl2\+σg2\+ρq2​G2\)\\displaystyle\+C\_\{1\}\\beta\_\{t\}\\left\(\\sigma\_\{l\}^\{2\}\+\\sigma\_\{g\}^\{2\}\+\\rho\_\{q\}^\{2\}G^\{2\}\\right\)\+C2​βt​L2​𝔼​\[‖𝐮t⋆‖2\]\.\\displaystyle\+C\_\{2\}\\beta\_\{t\}L^\{2\}\\mathbb\{E\}\\\!\\left\[\\left\\\|\\mathbf\{u\}\_\{t\}^\{\\star\}\\right\\\|^\{2\}\\right\]\.
By the SCM residual\-control condition in Assumption 5, the fixed\-point residual contribution satisfies

𝔼​\[⟨∇F​\(𝜽t\),𝐞SCM,t⟩\]≤βt4​𝔼​\[‖∇F​\(𝜽t\)‖2\]\+C3​RSCM,t2\.\\mathbb\{E\}\\\!\\left\[\\left\\langle\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\),\\mathbf\{e\}\_\{\\mathrm\{SCM\},t\}\\right\\rangle\\right\]\\leq\\frac\{\\beta\_\{t\}\}\{4\}\\mathbb\{E\}\\\!\\left\[\\left\\\|\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\)\\right\\\|^\{2\}\\right\]\+C\_\{3\}R\_\{\\mathrm\{SCM\},t\}^\{2\}\.\(37\)
Substituting \([35](https://arxiv.org/html/2609.00356#A1.E35)\), \([36](https://arxiv.org/html/2609.00356#A1.E36)\), and \([37](https://arxiv.org/html/2609.00356#A1.E37)\) into \([31](https://arxiv.org/html/2609.00356#A1.E31)\), and absorbing universal constants intoC1,C2,C3C\_\{1\},C\_\{2\},C\_\{3\}, yields

𝔼​\[F​\(𝜽t\+1\)\]≤\\displaystyle\\mathbb\{E\}\\\!\\left\[F\(\\boldsymbol\{\\theta\}\_\{t\+1\}\)\\right\]\\leq𝔼​\[F​\(𝜽t\)\]−βt2​𝔼​\[‖∇F​\(𝜽t\)‖2\]\\displaystyle\\mathbb\{E\}\\\!\\left\[F\(\\boldsymbol\{\\theta\}\_\{t\}\)\\right\]\-\\frac\{\\beta\_\{t\}\}\{2\}\\mathbb\{E\}\\\!\\left\[\\left\\\|\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\)\\right\\\|^\{2\}\\right\]\(38\)\+C1​βt​\(σl2\+σg2\+ρq2​G2\)\\displaystyle\+C\_\{1\}\\beta\_\{t\}\\left\(\\sigma\_\{l\}^\{2\}\+\\sigma\_\{g\}^\{2\}\+\\rho\_\{q\}^\{2\}G^\{2\}\\right\)\+C2​βt​L2​𝔼​\[‖𝐮t⋆‖2\]\+C3​RSCM,t2\.\\displaystyle\+C\_\{2\}\\beta\_\{t\}L^\{2\}\\mathbb\{E\}\\\!\\left\[\\left\\\|\\mathbf\{u\}\_\{t\}^\{\\star\}\\right\\\|^\{2\}\\right\]\+C\_\{3\}R\_\{\\mathrm\{SCM\},t\}^\{2\}\.
This proves Lemma[IV\.6](https://arxiv.org/html/2609.00356#S4.Thmtheorem6)\.

### A\-BProof of Theorem[IV\.7](https://arxiv.org/html/2609.00356#S4.Thmtheorem7)

For a constant geometry gainβt=β\\beta\_\{t\}=\\beta, summing \([38](https://arxiv.org/html/2609.00356#A1.E38)\) overt=0,…,T−1t=0,\\ldots,T\-1gives

𝔼​\[F​\(𝜽T\)\]≤\\displaystyle\\mathbb\{E\}\\left\[F\(\\boldsymbol\{\\theta\}\_\{T\}\)\\right\]\\leqF​\(𝜽0\)−β2​∑t=0T−1𝔼​\[‖∇F​\(𝜽t\)‖2\]\\displaystyle F\(\\boldsymbol\{\\theta\}\_\{0\}\)\-\\frac\{\\beta\}\{2\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\\left\[\\left\\\|\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\)\\right\\\|^\{2\}\\right\]\(39\)\+C1​β​T​\(σl2\+σg2\+ρq2​G2\)\\displaystyle\+C\_\{1\}\\beta T\\left\(\\sigma\_\{l\}^\{2\}\+\\sigma\_\{g\}^\{2\}\+\\rho\_\{q\}^\{2\}G^\{2\}\\right\)\+C2​β​L2​∑t=0T−1𝔼​\[‖𝐮t⋆‖2\]\+C3​∑t=0T−1RSCM,t2\.\\displaystyle\+C\_\{2\}\\beta L^\{2\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\\left\[\\left\\\|\\mathbf\{u\}\_\{t\}^\{\\star\}\\right\\\|^\{2\}\\right\]\+C\_\{3\}\\sum\_\{t=0\}^\{T\-1\}R\_\{\\mathrm\{SCM\},t\}^\{2\}\.
SinceFFis lower bounded byF⋆F^\{\\star\}, we have

𝔼​\[F​\(𝜽T\)\]≥F⋆\.\\mathbb\{E\}\\left\[F\(\\boldsymbol\{\\theta\}\_\{T\}\)\\right\]\\geq F^\{\\star\}\.Rearranging \([39](https://arxiv.org/html/2609.00356#A1.E39)\) and dividing byβ​T/2\\beta T/2yields

1T​∑t=0T−1𝔼​\[‖∇F​\(𝜽t\)‖2\]≤\\displaystyle\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\\left\[\\left\\\|\\nabla F\(\\boldsymbol\{\\theta\}\_\{t\}\)\\right\\\|^\{2\}\\right\]\\leq2​\(F​\(𝜽0\)−F⋆\)β​T\\displaystyle\\frac\{2\\left\(F\(\\boldsymbol\{\\theta\}\_\{0\}\)\-F^\{\\star\}\\right\)\}\{\\beta T\}\(40\)\+2​C1​\(σl2\+σg2\+ρq2​G2\)\\displaystyle\+2C\_\{1\}\\left\(\\sigma\_\{l\}^\{2\}\+\\sigma\_\{g\}^\{2\}\+\\rho\_\{q\}^\{2\}G^\{2\}\\right\)\+2​C2​L2T​∑t=0T−1𝔼​\[‖𝐮t⋆‖2\]\\displaystyle\+\\frac\{2C\_\{2\}L^\{2\}\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\\left\[\\left\\\|\\mathbf\{u\}\_\{t\}^\{\\star\}\\right\\\|^\{2\}\\right\]\+2​C3β​T​∑t=0T−1RSCM,t2\.\\displaystyle\+\\frac\{2C\_\{3\}\}\{\\beta T\}\\sum\_\{t=0\}^\{T\-1\}R\_\{\\mathrm\{SCM\},t\}^\{2\}\.
Absorbing fixed numerical factors intoC1,C2,C3C\_\{1\},C\_\{2\},C\_\{3\}, we obtain Eq\. \([29](https://arxiv.org/html/2609.00356#S4.E29)\)\.

If the SCM solver is run untilRSCM,t≤εSCMR\_\{\\mathrm\{SCM\},t\}\\leq\\varepsilon\_\{\\mathrm\{SCM\}\}for alltt, then

1T​∑t=0T−1RSCM,t2≤εSCM2\.\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}R\_\{\\mathrm\{SCM\},t\}^\{2\}\\leq\\varepsilon\_\{\\mathrm\{SCM\}\}^\{2\}\.Therefore, the residual contribution satisfies

C3β​T​∑t=0T−1RSCM,t2≤C3β​εSCM2,\\frac\{C\_\{3\}\}\{\\beta T\}\\sum\_\{t=0\}^\{T\-1\}R\_\{\\mathrm\{SCM\},t\}^\{2\}\\leq\\frac\{C\_\{3\}\}\{\\beta\}\\varepsilon\_\{\\mathrm\{SCM\}\}^\{2\},which gives

𝒪​\(εSCM2β\)\.\\mathcal\{O\}\\left\(\\frac\{\\varepsilon\_\{\\mathrm\{SCM\}\}^\{2\}\}\{\\beta\}\\right\)\.This yields the simplified convergence rate in Eq\. \([30](https://arxiv.org/html/2609.00356#S4.E30)\)\.