Fuzzy C-Means and Explainable AI for Quantum Entanglement Classification and Noise Analysis
Abstract
:1. Introduction
- Quantum Noise Modeling: Building on previous studies of quantum channels such as depolarization and phase noise [8], the effects of noise on Bell states are modeled, analyzing its impact on fidelity and entropy.
- Noise Suppression: Inspired by Steane’s work on error-correction codes and corrective operators [9], methods are implemented to restore the initial quantum state under different noise levels.
- Fuzzy and Hard Clustering: Models such as Fuzzy C-Means are applied [10] to classify quantum states based on fidelity and entropy, providing a probabilistic perspective on state transitions.
- XAI Integration: Techniques are incorporated such as SHAP [11] to explain the most influential variables in state recovery and cluster classification.
- Practical Case Study: Detailed simulations based on modern quantum-computing platforms (Qiskit v. 0.7.2, 2023) are conducted to evaluate the impact of noise and the effectiveness of corrective operators in preserving quantum entanglement and mitigating decoherence effects [12].
2. Related Work
2.1. Quantum Noise and Decoherence
- Depolarizing Channel: Introduces uniform noise by mixing the basis states with equal probabilities and has been widely used to model the degradation of entanglement in experimental systems [8]. Depolarizing noise is a worst-case scenario in which the quantum state completely loses its coherence and population information due to random errors. This type of noise is common in noisy intermediate-scale quantum (NISQ) devices and in situations where the system is subjected to uncontrolled interactions.
- Phase Noise: Models the loss of relative coherence between basis states, primarily affecting quantum interference [13]. Phase damping occurs when a quantum system loses coherence but retains population information, meaning that the state amplitudes remain unchanged, but superposition is degraded. This type of noise is less destructive than depolarization and appears in systems where phase information is lost due to environmental coupling without energy exchange.
- Amplitude damping: Simulates the loss of energy from an excited state to a bulk state, which is relevant in physical systems such as trapped atoms [14]. Amplitude damping is a significant type of quantum noise that simulates the loss of energy from an excited state to a bulk state. It is especially determinant in physical systems where quantum information is stored in energy levels, such as trapped ions, superconducting qubits and quantum dots.
2.2. Quantum Error Correction
2.3. Quantum-State Classification
2.4. XAI Techniques Applied to Quantum Systems
- SHAP: Assesses the global impact of individual variables within the classification model, emphasizing key metrics such as fidelity and entropy [16].
2.5. Quantum Teleportation in the Presence of Noise
2.6. Web of Science
- “Quantum noise AND entanglement”: Identifying articles that examine the impact of quantum noise on entangled systems.
- “Clustering AND quantum systems”: Searching for applications of Fuzzy C-Means in quantum-state classification.
- “Explainable AI AND quantum computing”: Identifying recent studies that integrate Explainable AI (XAI) into quantum analysis.
- “Quantum teleportation AND decoherence”: Reviewing both experimental and theoretical studies on quantum teleportation in the presence of noise.
2.6.1. Quantum Noise and Entanglement
2.6.2. Clustering and Quantum System
2.6.3. Explainable AI and Quantum Computing
3. Methodology
3.1. Initial Quantum-State Generation
- Quantum fidelity measures the “purity” of a state relative to an ideal reference state For identical states, fidelity is defined as:
- Von Neumann entropy evaluates the degree of disorder or uncertainty in a quantum system:
- Concurrence quantifies the degree of entanglement in a two-qubit system. For the Bell state , concurrence is maximized, .
3.2. Introduction of Quantum Noise
3.2.1. Depolarizing Channel
3.2.2. Damping Channel
3.2.3. Noise Impact Evaluation
3.3. Noise Suppression and State Recovery
3.3.1. Corrective Operator
- Depolarization: Transformations are applied to reinforce correlations between the system’s basis states.
- Phase Noise: Specific corrective operators realign the relative phases of the affected basis states.
3.3.2. Evaluation Metrics
3.4. Quantum-State Classification
3.5. Temporal Evaluation and Advanced Metrics
3.5.1. Temporal Evolution
3.5.2. Explainable Artificial Intelligence (XAI)
- Temporal Degradation Rate () measures how quickly fidelity declines due to noise accumulation over time:
- Corrective Recovery Rate () evaluates the effectiveness of noise suppression after applying a corrective operator:
4. Case Study
4.1. Initial Quantum-State Generation
- Hadamard gate () on Qubit 0: creates an equal superposition of and .
- CNOT gate () between Qubit 0 and Qubit 1: correlates the first qubit with the second, establishing quantum entanglement.
- A fidelity of 1 indicates that the generated state perfectly matches the ideal Bell state. This validates that the quantum circuit and the simulation work optimally in a noise-free environment.
- A value of 0 in the von Neumann entropy calculation confirms that the state is pure, which is a key feature of Bell states in an ideal environment.
- A value of 1 in the Concurrency metric indicates that the state is fully entangled, which is characteristic of Bell states.
4.2. Impact of Quantum Noise on Entropy and Fidelity
4.3. Noise Suppression and Effectiveness Analysis
- Depolarizing Noise Remains Highly Disruptive. Fidelity continued to decrease rapidly, even after correction, indicating that depolarizing noise introduces randomness that cannot be easily reversed. Entropy remained high, confirming that the quantum state was still highly mixed and unsuitable for computation.
- Phase Damping Demonstrates Greater Stability. Fidelity declined more gradually, suggesting that some coherence was retained post-noise. Entropy increased steadily but remained significantly lower than in depolarizing noise, meaning that the system retained more structure despite decoherence.
- Noise Suppression Had a Limited Effect. The corrective operator did not significantly alter fidelity or entropy trends. This suggests that simple corrective techniques are insufficient to counteract quantum noise, particularly in the case of depolarization. Phase damping proved more manageable, but even in this case, coherence recovery was not substantial.
4.4. Analysis of Clustering Results
- Cluster 1 (Centroid: Fidelity = 0.746, Entropy = 0.741). Represents states with moderate fidelity and entropy. These states retain some coherence but show signs of increased randomness. Could correspond to partially recoverable states after noise effects. Moderate-entropy states may still be recoverable, suggesting that targeted error-correction strategies could improve their usability.
- Cluster 2 (Centroid: Fidelity = 0.711, Entropy = 1.200). Shows the highest entropy, indicating states that are highly mixed and degraded. These states are likely not viable for quantum computation due to their low coherence. Associated with states that have undergone significant decoherence, possibly from depolarizing noise. High-entropy states are largely unusable, indicating that certain noise types (such as depolarization) have an irreversible impact on quantum coherence.
- Cluster 3 (Centroid: Fidelity = 0.739, Entropy = 0.264). Represents states with low entropy and moderate fidelity, suggesting minimal decoherence. These states remain the most stable and usable for quantum operations. Likely corresponds to states affected by milder noise sources, such as phase damping. Low-entropy states remain stable, implying that phase-damping effects may not completely degrade quantum information.
4.5. Temporal Evolution of Quantum States Under Noise and Correction
- Noise Level—the intensity of noise applied at each time step.
- Entropy—the level of uncertainty in the quantum state.
- Correction Applied—the magnitude of correction implemented to counteract noise effects.
- Time Step—the evolution of the quantum state over time.
- Temporal Fidelity Evolution Plot. A time-series graph that illustrates how fidelity changes over time when subjected to noise and correction. The confidence intervals (shaded region) represent the 95% confidence range for fidelity variations over multiple simulation runs. This plot helps determine whether correction strategies effectively preserve coherence or if the state remains irreversibly degraded; see Figure 11.
- SHAP Feature Importance Plot. The SHAP analysis of the model reveals that entropy is the most influential factor affecting quantum-state degradation, followed by noise level, time evolution, and the applied correction. These findings support the idea that classifying quantum states based on entropy and noise resistance is a more effective strategy than solely relying on noise suppression; see Figure 12.
- Entropy as the Dominant Factor Affecting Fidelity. The SHAP analysis clearly indicates that entropy has the highest impact on the model’s prediction of fidelity. This suggests that as quantum systems evolve under noise, entropy plays an important role in determining whether fidelity can be preserved. Higher entropy values correlate with a loss of coherence, making the system more prone to decoherence effects.
- Noise level is the second most influential factor. Noise level also has a significant impact but is less dominant than entropy. This implies that while noise is a major contributor to fidelity degradation, the way entropy accumulates over time is even more critical. This is consistent with the idea that quantum noise contributes to introducing randomness and fundamentally alters the structure of the system.
- Limited Impact of Correction Applied. The corrective measures applied have a relatively lower impact compared to entropy and noise. This suggests that the current correction techniques may not be sufficient to restore coherence effectively. It may be necessary to explore more sophisticated error-correction mechanisms, such as dynamically adapting the correction intensity based on real-time fidelity loss.
- Time-Step Influence is Minimal. The time-step factor has the least impact on fidelity, indicating that degradation due to noise is not necessarily linear over time. Instead, fluctuations in fidelity suggest that correction effects may vary unpredictably, potentially depending on other underlying interactions within the quantum system.
- Temporal Evolution of Fidelity Shows High Variability. The fidelity plot over time demonstrates a steady decline with oscillatory behavior. While some short-term recoveries are observed, the overall trend indicates a long-term irreversible loss of coherence. The error bands suggest significant variations in fidelity, possibly due to the stochastic nature of noise or variations in correction effectiveness.
5. Discussion and Future Work
5.1. Key Contributions and Novel Aspects
- The application of Fuzzy C-Means (FCM) for quantum-state classification: In contrast to previous research, which mainly focuses on direct noise suppression, clustering techniques are employed in the study to classify quantum states based on their resistance to noise.
- The integration of Explainable Artificial Intelligence (XAI) using SHAP analysis: By identifying the key factors influencing fidelity loss, an interpretable framework is provided to assess the degradation of quantum states. The results show that entropy has the largest impact, followed by noise level, correction intensity, and time evolution.
- Adaptive noise-mitigation strategies: SHAP enables the dynamic selection of corrective operators, optimizing error correction based on state-specific characteristics rather than predefined mechanisms. This improves efficiency in quantum error correction (QEC) by focusing on states that exhibit lower entropy and higher resilience to noise.
- Classification-driven approach to quantum-state management: Instead of applying uniform noise suppression, SHAP-guided classification enables the clustering of states by stability levels. This aligns with the Fuzzy C-Means (FCM) strategy employed in this study, offering a structured alternative to direct correction.
- An analysis of corrective operators and their limitations: The results indicate that standard correction techniques are ineffective against depolarizing noise, which shifts the paradigm towards classification and selective error mitigation.
- The identification of naturally resistant quantum states: Instead of attempting to correct all states, the proposed approach allows for the selection of states that naturally maintain computational feasibility despite noise.
5.2. Limitations of the Study
- Limited scope to two noise models: The noise models used in this study (depolarization and phase damping) represent standard theoretical frameworks but do not capture the full complexity of real-world quantum hardware noise, such as T1 relaxation, T2 dephasing, and crosstalk effects.
- Simplified correction operators: The correction techniques explored in this work are relatively simple and may not reflect more sophisticated quantum-error-correction (QEC) methods used in practical quantum computing.
- Experimental validation: The simulations are performed in a controlled computational environment, and real-world quantum hardware implementation is needed to validate these findings.
- Limited analysis of time evolution: Although the study explores the temporal behavior of quantum states after noise, longer time scales and different dynamical models could provide additional information.
- Fixed fuzziness parameter in FCM: The FCM clustering method was applied with a fixed fuzziness parameter , without a systematic sensitivity analysis. Further exploration is required to determine its optimal value in quantum-state clustering.
- Expanding quantum-state classification metrics: Fidelity and von Neumann entropy provide relevant information on quantum-state degradation, and alternative metrics such as quantum mutual information or distinguishability measures could further improve classification accuracy.
- The formalization of correction operators in quantum-error-correction frameworks: Although the impact of the correction operator on fidelity improvement has been empirically validated, a formal mathematical derivation of its structure within quantum-error-correction frameworks remains an open avenue for future research, potentially strengthening its theoretical foundation and applicability in practical quantum-computing scenarios.
- The scalability of SHAP in high-dimensional quantum systems: The computational cost of SHAP analysis scales with the complexity of the quantum system, as evaluating feature importance requires perturbing multiple input variables. While SHAP provided meaningful insights in our two-qubit experiments, its feasibility in larger Hilbert spaces remains an open question.
5.3. Future Work
- Extension to Additional Noise Models: Investigate how different types of quantum noise (e.g., amplitude damping, bit-flip, phase-flip) impact state classification and correction strategies.
- Refinement of Corrective Operators: Explore adaptive and optimized correction techniques, integrating machine learning to dynamically adjust correction strategies based on real-time state evolution.
- Hybrid Error-Mitigation Approaches: Combine clustering-based classification with quantum-error-correction (QEC) codes to enhance robustness against decoherence [28]. FCM-based classification can guide adaptive QEC strategies, applying minimal correction to resilient states while assigning stronger error correction, such as Steane or Surface Codes, to highly decohered states. SHAP-based interpretability further optimizes this process by dynamically selecting the most effective correction approach.
- Quantum Hardware Implementation: Validate the proposed methodology using real quantum processors (e.g., IBM Quantum, Rigetti) to assess its practical applicability. This will require adapting noise models to specific hardware characteristics, addressing measurement errors and optimizing the computational complexity of the clustering approach in high-dimensional quantum systems.
- The exploration of Reinforcement Learning for Noise Mitigation: Train reinforcement learning models to optimize correction intensity dynamically, improving state resilience.
- The development of Quantum-State Selection Strategies: Investigate methods for identifying and preparing naturally resilient entangled states that exhibit high stability under specific noise conditions.
- The optimization of FCM Clustering Parameters: Conduct a detailed study of the impact of the fuzziness parameter and different distance metrics on the classification of quantum states.
- The expansion of XAI Analysis: Extend the use of interpretability techniques beyond SHAP, incorporating additional explainability frameworks for a deeper understanding of noise impact. Future work should explore computationally efficient SHAP approximations or alternative XAI techniques tailored for high-dimensional quantum-state analysis.
- Extension to Multi-Qubit Systems and Complex Entangled States: Assess whether the proposed methodology remains effective for multi-qubit systems or alternative entangled, GHZ (Greenberger–Horne–Zeilinger) states [29] and W states [30]. This expansion could help identify generalizable noise patterns and improve corrective strategies in larger quantum networks.
- Integrating additional classification metrics beyond fidelity and entropy. Mutual quantum information could provide a more complete understanding of entanglement loss, while distinguishability measures, such as tracking distance, could refine the ability to differentiate quantum states under noise [31].
6. Conclusions
Funding
Data Availability Statement
Conflicts of Interest
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ClusterID | Fidelity | Entropy |
---|---|---|
Cluster 1 | 0.745822 | 0.740843 |
Cluster 2 | 0.710622 | 1.200312 |
Cluster 3 | 0.739229 | 0.263709 |
Id | Cluster 1 | Cluster 2 | Cluster 3 | Cluster | Fidelity | Entropy |
---|---|---|---|---|---|---|
47 | 0.98574 | 0.00625 | 0.00801 | Cluster 1 | 0.760034 | 0.703751 |
3 | 0.96800 | 0.01452 | 0.01748 | Cluster 1 | 0.799329 | 0.711999 |
63 | 0.95966 | 0.02183 | 0.01851 | Cluster 1 | 0.678377 | 0.750085 |
87 | 0.94384 | 0.03292 | 0.02324 | Cluster 1 | 0.818779 | 0.777281 |
36 | 0.92619 | 0.03637 | 0.03744 | Cluster 1 | 0.652307 | 0.726307 |
18 | 0.00935 | 0.98816 | 0.00250 | Cluster 2 | 0.715973 | 1.249583 |
27 | 0.01000 | 0.98756 | 0.00243 | Cluster 2 | 0.757117 | 1.205023 |
97 | 0.01149 | 0.98540 | 0.00311 | Cluster 2 | 0.713771 | 1.255954 |
65 | 0.02443 | 0.97009 | 0.00548 | Cluster 2 | 0.771348 | 1.169423 |
82 | 0.03339 | 0.95682 | 0.00979 | Cluster 2 | 0.665449 | 1.294571 |
17 | 0.00235 | 0.00061 | 0.99704 | Cluster 3 | 0.762378 | 0.261198 |
24 | 0.01754 | 0.00402 | 0.97843 | Cluster 3 | 0.728035 | 0.319109 |
89 | 0.03329 | 0.00725 | 0.95946 | Cluster 3 | 0.736107 | 0.338593 |
8 | 0.03697 | 0.00850 | 0.95453 | Cluster 3 | 0.800558 | 0.320317 |
23 | 0.04118 | 0.01309 | 0.94573 | Cluster 3 | 0.683181 | 0.154073 |
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Marín Díaz, G. Fuzzy C-Means and Explainable AI for Quantum Entanglement Classification and Noise Analysis. Mathematics 2025, 13, 1056. https://doi.org/10.3390/math13071056
Marín Díaz G. Fuzzy C-Means and Explainable AI for Quantum Entanglement Classification and Noise Analysis. Mathematics. 2025; 13(7):1056. https://doi.org/10.3390/math13071056
Chicago/Turabian StyleMarín Díaz, Gabriel. 2025. "Fuzzy C-Means and Explainable AI for Quantum Entanglement Classification and Noise Analysis" Mathematics 13, no. 7: 1056. https://doi.org/10.3390/math13071056
APA StyleMarín Díaz, G. (2025). Fuzzy C-Means and Explainable AI for Quantum Entanglement Classification and Noise Analysis. Mathematics, 13(7), 1056. https://doi.org/10.3390/math13071056