Noise-Resilient Quantum Federated Learning

• NR-QFL is a distributed quantum machine learning paradigm that fuses federated learning with tailored noise mitigation to counteract quantum hardware imperfections.
• It employs techniques such as sporadic update scaling, entropy-based client selection, and adaptive noise injection to ensure robust aggregation and privacy.
• Empirical results demonstrate improved accuracy, reduced communication load, and enhanced adversarial resilience even in high-noise quantum environments.

Noise-Resilient Quantum Federated Learning (NR-QFL) is a paradigm for distributed machine learning that integrates quantum computing and federated learning, with explicit mechanisms for mitigating the detrimental effects of quantum noise on training efficacy, convergence, and privacy. As contemporary quantum hardware is inherently noisy and heterogeneous, NR-QFL frameworks employ algorithmic, statistical, and architectural strategies to ensure scalable, robust, and provably secure collaboration among quantum-enabled clients.

1. Problem Formulation and Noise Modeling

NR-QFL considers a network of $K$ quantum clients, each with a local data distribution $D_k$ and quantum device characterized by a noise channel $\mathcal{N}_k$. The system seeks to optimize the following global objective: $$F(w) = \sum_{k=1}^K p_k\,F_k(w), \quad p_k \geq 0, \sum_k p_k = 1,$$ where

$$F_k(w) = \mathbb{E}_{(x, y) \sim D_k}\!\left[\ell\bigl(y, \mathrm{Tr}[O \cdot \mathcal{N}_k \circ U(w)(|0\rangle\langle0|)]\bigr)\right].$$

Here, $U(w)$ is a parameterized quantum circuit (PQC), $O$ is the measurement observable, and $\ell$ is a loss function such as cross-entropy (Rahman et al., 15 Jul 2025).

Quantum noise is modeled as a Completely-Positive Trace-Preserving (CPTP) map with Kraus operators $\{E_{k, j}\}$: $$\mathcal{N}_k(\rho) = \sum_j E_{k, j} \rho E_{k, j}^\dagger, \qquad \sum_j E_{k, j}^\dagger E_{k, j}=I.$$ The common case is the depolarizing channel: $$\mathcal{N}_k(\rho) = (1-\epsilon_k)\rho + \epsilon_k\,I/D.$$ Other standard channels include bit-flip, phase-flip, and amplitude-damping, each relevant for characterizing device-specific noise profiles (Sahu et al., 20 Jun 2024, Kabgere et al., 15 Dec 2025).

2. Noise-Resilience: Algorithmic Techniques

2.1 Sporadic Update Skipping and Scaling

In the SpoQFL framework, each client computes its noisy gradient $\widehat{g}_k^t = \nabla F_k(w_k^t) + \xi_k^t$ with $\mathrm{Var}(\xi_k^t) \leq V_k$. The local noise magnitude $\sigma_k(t) = |\xi_k^t|$ is used to define a sporadic variable $x_k^t = \exp(-\gamma \sigma_k(t))$. Updates for which $x_k^t$ falls below a threshold $\tau$ are either down-scaled or skipped: $$w_k^{t+1} = w_k^{t} - \eta\,\widehat{g}_k^t\,x_k^t\,, \qquad x_k^t \geq \tau.$$ The global aggregation weighs these sporadic factors: $$w^{t+1} = \sum_{k=1}^K \alpha_k(t) w_k^t, \quad \alpha_k(t) = \frac{p_k x_k^t}{\sum_j p_j x_j^t}$$ (Rahman et al., 15 Jul 2025).

2.2 Entropy-Based and Noise-Aware Client Selection

Entropy-based selection uses the von Neumann entropy $S(\rho_i) = -\mathrm{Tr}[\rho_i \log\rho_i]$ to assess the informativeness of each client’s quantum state. Jain’s fairness index is optimized to ensure balanced participation, suppressing outlier or adversarial updates and promoting robust aggregation across heterogeneous quantum hardware (Kabgere et al., 15 Dec 2025).

Noise-aware clustering and device selection, as in the NAC-QFL framework, operates by measuring device noise scores $\mathbb{N}_{\text{eff}}$ from calibration data (T1/T2, gate errors, readout errors, etc.), forming clusters minimizing intra-cluster communication and maximizing capacity, and selecting the subset with lowest cumulative $\mathbb{N}_{\text{eff}}$ such that resource and noise-threshold constraints are met. This enables deployment of smaller, higher-fidelity partitioned circuits on the best hardware (Sahu et al., 20 Jun 2024).

2.3 Adaptive and Differential-Privacy Noise Injection

Noise is utilized both for DP purposes and to restore gradient variance in quantum neural networks (addressing vanishing-gradient phenomena, e.g., barren plateaus). Adaptive Gaussian noise is injected per-client, per-round, with variance schedule $\sigma_t^2 = \sigma_0^2/(1+\alpha t)$ calibrated via the DP budget: $$\sigma_0^2 \geq \frac{8\,T\,(2L+b)^2\log(1/\delta)}{K^2\epsilon^2}$$ where $K$ is the per-round client count, $L$ is the smoothness parameter, and $T$ the number of rounds (Phan et al., 4 Sep 2025, Pokharel et al., 27 Aug 2025).

Quantum noise from measurement shot noise and gate depolarization (with total depolarizing factor $p = 1 - (1-\lambda)^L$) can be leveraged for DP, with the privacy parameter: $$\varepsilon_{n, t} = \frac{\Delta}{\sigma}\sqrt{2\ln(1.25/\delta)}$$ where $\Delta$ is the $\ell_2$ sensitivity of the clipped gradient, and $\sigma$ is set by both device-induced and intentionally added noise (Pokharel et al., 27 Aug 2025).

3. Aggregation Structures and Quantum Protocols

3.1 Quantum State Encoding and Variational Aggregation

Model weights are mapped to quantum states via angle encoding: $$|\psi_i\rangle = R_y(w_i)|0\rangle = \cos\left(\frac{w_i}{2}\right)|0\rangle + \sin\left(\frac{w_i}{2}\right)|1\rangle$$ for each scalar parameter, extended to multi-qubit product states for full models.

A variational quantum circuit (VQC) on the server acts as $U_{\rm agg}$ over the entangled register, with gates parameterized adaptively for noise compensation. Aggregated results are extracted via projective measurements and mapped back to classical weights by estimating $\langle Z_j \rangle$ for each qubit and inverting the encoding (Kabgere et al., 15 Dec 2025).

3.2 Circuit Partitioning and Multi-Server Orchestration

Partitioning complex quantum circuits into smaller subcircuits (“circuit cutting”) enables execution on low-noise devices with limited qubit resources, improving overall fidelity. In multi-server settings, the model is divided into $M$ blocks, each aggregated by an independent server, and later reassembled classically; this design enhances fault-tolerance and minimizes both communication and aggregation bottlenecks (Sahu et al., 20 Jun 2024, Kabgere et al., 15 Dec 2025).

4. Convergence, Error Bounds, and Robustness Guarantees

NR-QFL frameworks provide explicit convergence and steady-state error guarantees under noise:

• Sporadic scaling yields improved noise bounds: If $E[x_k^t{}^2] < 1$, the effective variance $V_{\rm sp} = \sum_k p_k V_k E[x_k^t{}^2]$ tightens the error bound:

$$\mathbb{E}[F(w^T)]-F(w^*) \leq (1-\eta\mu)^T(F(w^0)-F(w^*)) + \frac{\eta L V_{\rm sp}}{2\mu}$$

(Rahman et al., 15 Jul 2025).

• Trace-distance and variance scaling: For quantum aggregation on NISQ devices, the error in the aggregated quantum state scales linearly with noise rate and circuit depth ($\epsilon = O(p\,L)$), and the variance in estimated parameters tightens as $O(1/N)$ with the number of clients (Kabgere et al., 15 Dec 2025).
• Gradient variance and DP regularization: Adaptive noise replenishes lost variance due to quantum measurement-induced suppression in deep circuits, mitigating barren-plateau effects; the variance contribution vanishes exponentially with qubit count, but adaptive injection preserves sufficient variance for effective, stable training (Phan et al., 4 Sep 2025).
• Adversarial robustness: NR-QFL frameworks such as RobQFL explicitly train a tunable fraction $\gamma$ of clients under adversarial perturbations, optimizing trade-offs via fixed or mixed $\epsilon$ schedules. Robustness metrics such as ARA and RV quantitatively capture the system’s resilience to attacks and noise, exposing the severe impact of non-IID data (Maouaki et al., 5 Sep 2025).

5. Experimental Results and Practical Impact

Empirical evaluations demonstrate the effectiveness of NR-QFL across multiple noisy quantum regimes and learning tasks:

Framework	Dataset	Noise/DP Strength	Accuracy (%)	Notable Gains
SpoQFL (Rahman et al., 15 Jul 2025)	CIFAR-10	$\epsilon=0.001$	91.92	+4.87 vs wpQFL
	CIFAR-100	$\epsilon=0.001$	57.60	+3.66 vs wpQFL
NAC-QFL (Sahu et al., 20 Jun 2024)	MNIST-Bin	$p=0.05$	up to 97.5	$+4$ vs naive FL
NR-QFL ADAS (Kabgere et al., 15 Dec 2025)	CIFAR-10	$p=0.05$	86.1	$+6.9$ vs FedAvg
DP-QFL (Pokharel et al., 27 Aug 2025, Phan et al., 4 Sep 2025)	MNIST	$\varepsilon\leq 10$	87.6	$+7$–$20$ adv. rob.
	CIFAR-10	$\varepsilon\leq 10$	79.9	DP/rob. tradeoff

Key observations include:

• NR-QFL frameworks sustain high accuracy ($>85\%$ on CIFAR-10) even at depolarizing noise rates up to $p=0.08$ (Kabgere et al., 15 Dec 2025).
• Communication cost is consistently reduced by targeting small, low-noise subcircuits and a small set of selected clients (Sahu et al., 20 Jun 2024).
• Differential-privacy integration via quantum noise secures $(\varepsilon, \delta)$-DP budgets of $\sim 5$–10 with minimal performance loss (Pokharel et al., 27 Aug 2025, Phan et al., 4 Sep 2025).
• Adversarial resilience is substantially improved by mixing adversarial and clean training; the RV and ARA metrics reveal that moderate adversarial coverage ($\gamma = 20$–$50\%$) delivers $\sim$15 p.p. gains in robustness at negligible clean-accuracy cost (Maouaki et al., 5 Sep 2025).
• The convergence of NR-QFL is empirically faster (e.g., $>90\%$ accuracy in 300 rounds vs. 400–500 rounds for baselines) (Phan et al., 4 Sep 2025).

6. Limitations and Future Research Directions

Two primary sources of limitation are highlighted across NR-QFL literature:

• Client/data heterogeneity: Non-IID distributions (e.g., label-sorted splits) halve robustness and elevate aggregation conflicts (Maouaki et al., 5 Sep 2025). Mitigation requires introducing public proxy datasets, personalized aggregation, or dynamic curriculum schedules.
• Device/resource constraints: Device selection under noise/capacity and circuit partitioning is NP-hard; heuristics and parallelism caps are necessary when federating on a large hardware pool (Sahu et al., 20 Jun 2024).
• Scalability/Bottlenecks: Classical post-processing for circuit cutting and the orchestration of uncertainty in large, heterogeneous systems remain open avenues for optimization (Kabgere et al., 15 Dec 2025).

A plausible implication is that NR-QFL will increasingly combine quantum-aware error mitigation (PEC/ZNE), differentially-private aggregation, and adversarially-tuned optimization to maintain robust performance as both quantum hardware and data scale—a requirement for ADAS, mobile networks, and other safety-critical distributed AI applications.

• "Sporadic Federated Learning Approach in Quantum Environment to Tackle Quantum Noise" (Rahman et al., 15 Jul 2025)
• "RobQFL: Robust Quantum Federated Learning in Adversarial Environment" (Maouaki et al., 5 Sep 2025)
• "Noise-Resilient Quantum Aggregation on NISQ for Federated ADAS Learning" (Kabgere et al., 15 Dec 2025)
• "Differentially Private Federated Quantum Learning via Quantum Noise" (Pokharel et al., 27 Aug 2025)
• "Enhancing Gradient Variance and Differential Privacy in Quantum Federated Learning" (Phan et al., 4 Sep 2025)
• "NAC-QFL: Noise Aware Clustered Quantum Federated Learning" (Sahu et al., 20 Jun 2024)