Fault-Injection Probing: A Causal Interpretability Framework for Quantum Machine Learning Models

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Sundaram¹ This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8766996/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Quantum machine learning (QML) models achieve competitive performance on real-world tasks, yet interpreting what these models learn remains an open challenge. Classical interpretability techniques depend on access to intermediate representations, which quantum systems forbid due to measurement collapse, the no-cloning theorem, and exponential state-space dimensionality. We introduce Fault-Injection Probing (FIP), a framework that repurposes controlled quantum errors—bit flips, phase flips, depolarising channels, and erasure—as interpretability probes. FIP injects a known fault at a specific qubit and circuit layer, then measures the output shift. Comparing shifts across inputs with and without a target feature yields causal attribution scores linking qubits to learned representations. On variational quantum classifiers trained for sentiment analysis, FIP identifies sentiment-encoding qubits whose targeted perturbation flips 72% of relevant predictions. On a synthetic benchmark with known ground-truth mappings, FIP achieves 100% identification accuracy with zero false positives across all eight qubits. The framework is model-agnostic, extending to quantum kernels, reservoir models, and QAOA, and supports practical applications including model debugging and adversarial robustness assessment. Quantum machine learning Interpretability Explainable AI Fault injection Causal inference 1. Introduction Quantum machine learning has moved beyond theoretical curiosity. Variational quantum classifiers now achieve competitive accuracy on standard benchmarks [ 1 ]. Quantum kernels provide provable computational advantages for certain problem classes [ 2 ]. Reservoir-based quantum models sidestep training difficulties that plague parameterised circuits [ 3 ]. However, a fundamental question persists: what do these quantum models actually learn? Classical deep learning offers a rich interpretability toolkit [ 4 ]. Gradient-based saliency maps highlight influential input features. Attention visualisations expose token-level relationships. Probing classifiers test what information hidden layers encode. Perturbation methods such as LIME [ 9 ] and SHAP [ 10 ] measure local sensitivity. All these techniques share a common prerequisite: access to intermediate representations. Quantum systems violate this prerequisite for reasons rooted in physics. Measurement collapse destroys superpositions mid-computation. The no-cloning theorem [ 5 ] forbids state duplication. An n-qubit system occupies a 2ⁿ-dimensional Hilbert space, making full characterisation intractable. Complex phases remain invisible to standard measurement yet encode distinct information. These constraints mean that closing the quantum interpretability gap requires a fundamentally different approach. We observe that while direct observation of quantum states is forbidden, intervention remains possible. Quantum errors—the phenomena that limit quantum computing reliability—become powerful diagnostic instruments when applied deliberately. Fault-Injection Probing (FIP) inserts controlled faults at targeted circuit locations and measures how the output distribution changes. Interventions establish cause-and-effect relationships, grounding FIP in causal inference [ 16 ]. The method respects quantum constraints by observing only final measurement outcomes. Fault operations use standard quantum gates (Pauli-X, Pauli-Z, depolarising channels, qubit resets), requiring no exotic hardware. Probing different qubits, layers, and fault types produces a granular picture of internal information organisation. We demonstrate that FIP reliably identifies feature-encoding qubits in trained classifiers. On sentiment analysis circuits, it pinpoints sentiment-carrying qubits. On a synthetic benchmark with known ground truth, it achieves perfect accuracy. The framework generalises across variational circuits, quantum kernels, and reservoir models. 2. Related work Gradient-based attribution methods—saliency maps [ 6 ], integrated gradients [ 7 ], Grad-CAM [ 8 ]—trace input feature influence via gradient flow. They require differentiable forward passes and efficient gradient computation. For quantum circuits, gradient estimation via the parameter-shift rule demands exponentially many circuit executions at scale. Perturbation-based methods (LIME [ 9 ], SHAP [ 10 ]) perturb inputs and fit local surrogate models. These translate to quantum systems in principle but reveal nothing about internal circuit structure. Probing classifiers [ 11 ] train auxiliary models on intermediate representations, which are inaccessible in quantum circuits. Quantum state tomography [ 12 ] reconstructs the full density matrix but is exponentially expensive and yields raw state data rather than functional interpretation. Noise characterisation [ 13 ] treats noise as a problem; we repurpose it as a tool. Quantum feature map expressiveness analysis [ 14 ] addresses model capacity, not learned representations. Barren plateau studies [ 15 ] illuminate training landscapes without revealing what models encode. To our knowledge, no prior work offers a systematic method for interpreting trained quantum circuits. 3. Fault-Injection Probing framework 3.1. Problem formulation We consider a trained quantum classifier f_θ: X → Y, where X is the input space, Y contains class probability distributions, and θ collects trained parameters. The classifier has three stages: an encoding circuit U_enc(x) mapping classical inputs to quantum states, a variational circuit U_var(θ) applying learned transformations, and measurement extracting class probabilities. Our goal is to determine which qubits encode which features, how information flows through layers, and which components are critical for specific predictions. 3.2. Fault models We define four fault types, each probing a different aspect of quantum information. The bit-flip fault inserts a Pauli-X gate at qubit i after layer l, effecting ρ → X_iρX_i. This flips the computational basis state and probes classical-like information. The phase-flip fault inserts a Pauli-Z gate (ρ → Z_iρZ_i), targeting quantum coherence information. The depolarising fault applies a depolarising channel with probability p, partially erasing information to probe encoding robustness. The erasure fault resets qubit i to |0⟩, completely destroying its information content to test necessity. 3.3. Sensitivity metrics Given normal output y = f_θ(x) and faulted output y' = f_θ^{(i,l,F)}(x), we define four metrics. L2 sensitivity captures the overall output shift magnitude: S_L2 = ||y − y'||_2. Prediction flip indicates whether the fault changes the predicted class. Confidence change measures certainty shift even when the top prediction is unchanged. KL sensitivity computes the Kullback–Leibler divergence between original and faulted distributions. 3.4. Feature attribution We aggregate sensitivities over a dataset D and split by a binary feature φ. The feature attribution score is A(i, φ, F) = S̄_{φ = 1}(i,F) − S̄_{φ = 0}(i,F). A positive score indicates that qubit i is more sensitive when the input contains feature φ, providing evidence that the qubit encodes that feature. 3.5. Statistical framework We test whether each qubit's sensitivity depends on feature φ using a two-sample t-test. Bonferroni correction sets the threshold to α/n_qubits. We also report Cohen's d as an effect size measure. A qubit is flagged as feature-encoding only when both the corrected p-value is significant and d exceeds 0.5. This dual criterion guards against statistical noise and trivially small effects. 4. Algorithms Algorithm 1 (Fault-Injection Probing): Given a trained classifier, probe dataset D, fault types F, and probe points P (qubit-layer pairs), the algorithm iterates over all inputs, probe points, and faults. For each combination, it injects the fault, executes the circuit, and records the sensitivity. The output is a sensitivity tensor S[input, probe_point, fault_type] with aggregated statistics. Complexity scales as O(|D| × |P| × |F| × circuit_cost). All circuit executions are independent and trivially parallelisable. Algorithm 2 (Feature Attribution): For each qubit, sensitivity values are collected separately for inputs with (φ = 1) and without (φ = 0) the target feature. The attribution score is the difference in mean sensitivity. Statistical significance is assessed via t-test with Bonferroni correction. Algorithm 3 (Causal Validation): For a qubit hypothesised to encode a feature, the algorithm computes prediction flip rates under fault injection for positive and negative examples. Validation succeeds when the positive flip rate exceeds 50% and is at least twice the negative flip rate. This dual criterion ensures the qubit genuinely encodes the feature rather than being generically fragile. 5. Experimental validation 5.1. Experimental setup We evaluate FIP on three architectures: a variational quantum classifier (VQC) with 8–12 qubits and 4–6 layers, a quantum reservoir model (QRLM) with 12 qubits, and a quantum kernel classifier with 10 qubits. Datasets include SST-2 (1,000 sentiment examples), AG News (1,000 topic examples), and a synthetic dataset (500 examples) with known ground-truth feature–qubit mappings. All circuits are simulated using PennyLane with 1,024 shots per execution. Results are averaged over five random seeds. 5.2. Sentiment classification We trained a VQC on SST-2 binary sentiment and applied FIP to identify sentiment-encoding qubits. Table 1 presents the bit-flip probing results. Table 1 Bit-flip probing results for sentiment classification on SST-2. Qubit Sens. (Pos) Sens. (Neg) Attribution p-value Q3 0.42 0.11 + 0.31 0.0001 Q7 0.38 0.09 + 0.29 0.0003 Q1 0.15 0.14 + 0.01 0.847 Q5 0.18 0.16 + 0.02 0.612 Qubits 3 and 7 show sensitivity three to four times higher for positive-sentiment inputs. Both p-values survive Bonferroni correction. Qubits 1 and 5 show nearly identical sensitivity regardless of sentiment, suggesting they carry non-sentiment information. Table 2 presents the causal validation results through targeted interventions. Table 2 Validation through targeted intervention on sentiment-encoding qubit candidates. Intervention Flip Rate (Pos) Flip Rate (Neg) Validated Bit flip Q3 72% 23% Yes Bit flip Q7 68% 19% Yes Bit flip Q1 31% 28% No Phase flip Q3 45% 18% Yes Flipping qubit 3 changes predictions for 72% of positive-sentiment inputs but only 23% of negative ones. This three-to-one ratio provides strong causal evidence. A phase flip on Q3 also validates (45% vs. 18%), suggesting sentiment is encoded in both computational basis and quantum phase. 5.3. Synthetic ground truth To confirm FIP does not detect artefacts, we constructed a synthetic dataset where feature φ directly affects qubits 2 and 5 during encoding. Table 3 presents the results. Table 3 FIP attribution on synthetic dataset with known ground-truth mappings. Qubit True Encoding FIP Attribution Correct Q0 No 0.02 Yes Q1 No 0.01 Yes Q2 Yes 0.34 Yes Q3 No 0.03 Yes Q4 No 0.01 Yes Q5 Yes 0.41 Yes Q6 No 0.05 Yes Q7 No 0.02 Yes FIP achieves 100% identification accuracy. The lowest true-positive score (0.34) is nearly seven times the highest false-positive score (0.05). This clean separation confirms that the sentiment results are not spurious. 5.4. Layer-wise information flow FIP can track information flow through successive circuit layers. Table 4 shows sensitivity evolution at four circuit points. Table 4 Layer-wise sensitivity evolution for sentiment-encoding qubits. Layer Q3 Sens. Q7 Sens. Interpretation Post-encoding 0.15 0.12 Feature not yet concentrated Layer 2 0.28 0.25 Feature emerging Layer 4 0.42 0.38 Feature concentrated Pre-measurement 0.45 0.41 Stable representation What emerges is a concentration effect. Sentiment information starts distributed and progressively concentrates onto Q3 and Q7 through variational layers. By layer 4, the representation stabilises. This pattern mirrors classical transformers, where semantic features crystallise in middle-to-late layers. 5.5. Comparing fault types Table 5 compares the four fault models across our experimental suite. Table 5 Comparative analysis of fault types. Fault Type Avg Sensitivity Best for Detecting Bit flip 0.34 Classical-like information Phase flip 0.22 Coherence-encoded features Depolarise (p = 0.1) 0.28 Information robustness Erasure 0.48 Information necessity Erasure faults produce the strongest signal by completely destroying qubit information. Bit flips serve as the best general-purpose probe. Phase flips uniquely detect quantum-native features encoded in coherence rather than population. 5.6. Computational cost Table 6 reports wall-time measurements from our simulation environment. Table 6 Computational cost scaling for FIP. Qubits Probe Points Total Circuits Wall Time 8 32 6,400 12 min 12 48 9,600 28 min 16 64 12,800 51 min 20 80 16,000 89 min Cost scales linearly with qubits and probe points. An 8-qubit analysis finishes in under 15 minutes. Circuit executions are independent, enabling 4× speedup on a modest GPU cluster. 6. Applications FIP supports four practical application areas. For model debugging, practitioners identify failing inputs, run FIP to locate qubits with anomalous sensitivity patterns, and diagnose root causes (encoding capacity, optimisation failures, or architecture limits). For architecture search, FIP characterises internal representations of competing designs; architectures concentrating features on fewer qubits with higher attribution tend to generalise better. For trust and deployment in high-stakes settings, FIP maps predictions to feature-encoding qubits, enabling auditability. For adversarial robustness, FIP identifies critical qubits with high sensitivity and low redundancy, estimating vulnerability to targeted quantum noise attacks without explicit attack construction. 7. Discussion 7.1. Why FIP works FIP succeeds for four reasons. First, interventions establish causal links, not correlations. Second, quantum faults are natural operations—we use the physics rather than fighting it. Third, we measure only final outputs, so collapse is not an obstacle. Fourth, differential comparison between feature-present and feature-absent inputs isolates specific contributions. 7.2. Limitations FIP has notable limitations. Single-qubit probing may miss highly distributed, entangled encodings; multi-qubit coordinated probing addresses this at higher combinatorial cost. Hardware noise may mask injected faults on real devices; relative comparisons and increased shots mitigate this. Computational cost grows with dataset size, probe points, and fault types; adaptive sampling reduces overhead. Finally, FIP requires specifying features of interest rather than discovering them autonomously. 7.3. Connection to causal inference FIP is grounded in Pearl's interventionist causal framework [ 16 ]. Fault injections correspond to do-operations: P(Y|do(X_i = x')) rather than P(Y|X_i = x'). Interventions break confounding, ensuring FIP provides causal attribution rather than associative evidence. 7.4. Extension to other architectures Table 7 summarises FIP adaptations across quantum model types. Table 7 FIP adaptation across quantum model architectures. Model Type Adaptation Required VQC Direct application to variational layers Quantum Kernel Probe the feature map circuit Reservoir Model Probe reservoir and readout separately QAOA Probe mixer and cost layers independently The reservoir model case is particularly informative. Probing the fixed reservoir reveals inductive bias, while probing the readout reveals what training extracted. 8. Conclusion We introduced Fault-Injection Probing, a framework for interpreting trained quantum ML models using controlled quantum errors as diagnostic probes. FIP is grounded in causal inference theory and respects quantum mechanical constraints. On sentiment classification, it identifies sentiment-encoding qubits validated through targeted interventions. On synthetic ground truth, it achieves perfect attribution accuracy. FIP also enables layer-wise information flow tracking, fault-type comparison, model debugging, and adversarial robustness assessment. The framework is model-agnostic, extending to variational classifiers, quantum kernels, reservoir models, and QAOA. Limitations remain around distributed representations and computational scaling, but FIP provides a validated methodology where none existed before. As quantum models approach real-world deployment, the ability to interpret their learned representations will be essential for trust and reliability. Declarations Author Contributions Prabakaran Kannan: Conceptualization, Methodology, Software, Validation, Formal Analysis, Investigation, Data Curation, Writing – Original Draft, Visualization. Venkatesan M. Sundaram: Supervision, Writing – Review & Editing, Project Administration. Conflict of Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Ethics Statement This study is purely computational and involves no human participants, animal subjects, or personal data. All experiments were conducted using quantum circuit simulations on publicly available benchmark datasets. No ethical approval was required. Clinical trial number: not applicable. Data Availability The code and experimental data will be made available on GitHub upon acceptance. All benchmark datasets used (SST-2, AG News) are publicly available. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References Schuld M, Petruccione F (2018) Supervised Learning with Quantum Computers. Springer, Cham. Liu Y, Arunachalam S, Temme K (2021) A rigorous and robust quantum speed-up in supervised machine learning. Nature Physics 17:1013–1017. Fujii K, Nakajima K (2017) Harnessing disordered-ensemble quantum dynamics for machine learning. Physical Review Applied 8:024030. Lipton ZC (2018) The mythos of model interpretability. Queue 16(3):31–57. Wootters WK, Zurek WH (1982) A single quantum cannot be cloned. Nature 299:802–803. Simonyan K, Vedaldi A, Zisserman A (2014) Deep inside convolutional networks: visualising image classification models and saliency maps. In: ICLR Workshop. Sundararajan M, Taly A, Yan Q (2017) Axiomatic attribution for deep networks. In: Proceedings of ICML. Selvaraju RR, Cogswell M, Das A, Vedantam R, Parikh D, Batra D (2017) Grad-CAM: visual explanations from deep networks via gradient-based localization. In: Proceedings of ICCV. Ribeiro MT, Singh S, Guestrin C (2016) Why should I trust you?: explaining the predictions of any classifier. In: Proceedings of KDD. Lundberg SM, Lee SI (2017) A unified approach to interpreting model predictions. In: Proceedings of NeurIPS. Belinkov Y, Glass J (2019) Analysis methods in neural language processing: a survey. Transactions of the Association for Computational Linguistics 7:49–72. Paris M, Řeháček J (2004) Quantum State Estimation. Springer, Berlin. Nielsen MA, Chuang IL (2010) Quantum Computation and Quantum Information, 10th Anniversary Edition. Cambridge University Press, Cambridge. Schuld M, Sweke R, Meyer JJ (2021) Effect of data encoding on the expressive power of variational quantum machine-learning models. Physical Review A 103:032430. McClean JR, Boixo S, Smelyanskiy VN, Babbush R, Neven H (2018) Barren plateaus in quantum neural network training landscapes. Nature Communications 9:4812. Pearl J (2009) Causality: Models, Reasoning, and Inference, 2nd edn. Cambridge University Press, Cambridge. 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Sundaram¹","email":"","orcid":"","institution":"National Institute of Technology Puducherry","correspondingAuthor":false,"prefix":"","firstName":"Venkatesan","middleName":"M.","lastName":"Sundaram¹","suffix":""}],"badges":[],"createdAt":"2026-02-02 15:55:34","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8766996/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8766996/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":102297195,"identity":"61f454ed-2247-48b7-a323-d7ffea469494","added_by":"auto","created_at":"2026-02-10 10:26:25","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":716459,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8766996/v1/7ece2d14-5a88-433b-9f63-0dba6a7ef7ba.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Fault-Injection Probing: A Causal Interpretability Framework for Quantum Machine Learning Models","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eQuantum machine learning has moved beyond theoretical curiosity. Variational quantum classifiers now achieve competitive accuracy on standard benchmarks [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. Quantum kernels provide provable computational advantages for certain problem classes [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. Reservoir-based quantum models sidestep training difficulties that plague parameterised circuits [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. However, a fundamental question persists: what do these quantum models actually learn?\u003c/p\u003e \u003cp\u003eClassical deep learning offers a rich interpretability toolkit [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. Gradient-based saliency maps highlight influential input features. Attention visualisations expose token-level relationships. Probing classifiers test what information hidden layers encode. Perturbation methods such as LIME [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e] and SHAP [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e] measure local sensitivity. All these techniques share a common prerequisite: access to intermediate representations.\u003c/p\u003e \u003cp\u003eQuantum systems violate this prerequisite for reasons rooted in physics. Measurement collapse destroys superpositions mid-computation. The no-cloning theorem [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e] forbids state duplication. An n-qubit system occupies a 2ⁿ-dimensional Hilbert space, making full characterisation intractable. Complex phases remain invisible to standard measurement yet encode distinct information. These constraints mean that closing the quantum interpretability gap requires a fundamentally different approach.\u003c/p\u003e \u003cp\u003eWe observe that while direct observation of quantum states is forbidden, intervention remains possible. Quantum errors\u0026mdash;the phenomena that limit quantum computing reliability\u0026mdash;become powerful diagnostic instruments when applied deliberately. Fault-Injection Probing (FIP) inserts controlled faults at targeted circuit locations and measures how the output distribution changes. Interventions establish cause-and-effect relationships, grounding FIP in causal inference [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. The method respects quantum constraints by observing only final measurement outcomes. Fault operations use standard quantum gates (Pauli-X, Pauli-Z, depolarising channels, qubit resets), requiring no exotic hardware. Probing different qubits, layers, and fault types produces a granular picture of internal information organisation.\u003c/p\u003e \u003cp\u003eWe demonstrate that FIP reliably identifies feature-encoding qubits in trained classifiers. On sentiment analysis circuits, it pinpoints sentiment-carrying qubits. On a synthetic benchmark with known ground truth, it achieves perfect accuracy. The framework generalises across variational circuits, quantum kernels, and reservoir models.\u003c/p\u003e"},{"header":"2. Related work","content":"\u003cp\u003eGradient-based attribution methods\u0026mdash;saliency maps [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e], integrated gradients [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e], Grad-CAM [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]\u0026mdash;trace input feature influence via gradient flow. They require differentiable forward passes and efficient gradient computation. For quantum circuits, gradient estimation via the parameter-shift rule demands exponentially many circuit executions at scale. Perturbation-based methods (LIME [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e], SHAP [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]) perturb inputs and fit local surrogate models. These translate to quantum systems in principle but reveal nothing about internal circuit structure. Probing classifiers [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e] train auxiliary models on intermediate representations, which are inaccessible in quantum circuits.\u003c/p\u003e \u003cp\u003eQuantum state tomography [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e] reconstructs the full density matrix but is exponentially expensive and yields raw state data rather than functional interpretation. Noise characterisation [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e] treats noise as a problem; we repurpose it as a tool. Quantum feature map expressiveness analysis [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] addresses model capacity, not learned representations. Barren plateau studies [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] illuminate training landscapes without revealing what models encode. To our knowledge, no prior work offers a systematic method for interpreting trained quantum circuits.\u003c/p\u003e"},{"header":"3. Fault-Injection Probing framework","content":"\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1. Problem formulation\u003c/h2\u003e \u003cp\u003eWe consider a trained quantum classifier f_θ: X \u0026rarr; Y, where X is the input space, Y contains class probability distributions, and θ collects trained parameters. The classifier has three stages: an encoding circuit U_enc(x) mapping classical inputs to quantum states, a variational circuit U_var(θ) applying learned transformations, and measurement extracting class probabilities. Our goal is to determine which qubits encode which features, how information flows through layers, and which components are critical for specific predictions.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e3.2. Fault models\u003c/h2\u003e \u003cp\u003eWe define four fault types, each probing a different aspect of quantum information. The bit-flip fault inserts a Pauli-X gate at qubit i after layer l, effecting ρ \u0026rarr; X_iρX_i. This flips the computational basis state and probes classical-like information. The phase-flip fault inserts a Pauli-Z gate (ρ \u0026rarr; Z_iρZ_i), targeting quantum coherence information. The depolarising fault applies a depolarising channel with probability p, partially erasing information to probe encoding robustness. The erasure fault resets qubit i to |0⟩, completely destroying its information content to test necessity.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e3.3. Sensitivity metrics\u003c/h2\u003e \u003cp\u003eGiven normal output y\u0026thinsp;=\u0026thinsp;f_θ(x) and faulted output y' = f_θ^{(i,l,F)}(x), we define four metrics. L2 sensitivity captures the overall output shift magnitude: S_L2 = ||y\u0026thinsp;\u0026minus;\u0026thinsp;y'||_2. Prediction flip indicates whether the fault changes the predicted class. Confidence change measures certainty shift even when the top prediction is unchanged. KL sensitivity computes the Kullback\u0026ndash;Leibler divergence between original and faulted distributions.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e3.4. Feature attribution\u003c/h2\u003e \u003cp\u003eWe aggregate sensitivities over a dataset D and split by a binary feature φ. The feature attribution score is A(i, φ, F) = S̄_{φ\u0026thinsp;=\u0026thinsp;1}(i,F) \u0026minus; S̄_{φ\u0026thinsp;=\u0026thinsp;0}(i,F). A positive score indicates that qubit i is more sensitive when the input contains feature φ, providing evidence that the qubit encodes that feature.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.5. Statistical framework\u003c/h2\u003e \u003cp\u003eWe test whether each qubit's sensitivity depends on feature φ using a two-sample t-test. Bonferroni correction sets the threshold to α/n_qubits. We also report Cohen's d as an effect size measure. A qubit is flagged as feature-encoding only when both the corrected p-value is significant and d exceeds 0.5. This dual criterion guards against statistical noise and trivially small effects.\u003c/p\u003e \u003c/div\u003e"},{"header":"4. Algorithms","content":"\u003cp\u003e \u003cstrong\u003eAlgorithm 1\u003c/strong\u003e \u003cp\u003e(Fault-Injection Probing): Given a trained classifier, probe dataset D, fault types F, and probe points P (qubit-layer pairs), the algorithm iterates over all inputs, probe points, and faults. For each combination, it injects the fault, executes the circuit, and records the sensitivity. The output is a sensitivity tensor S[input, probe_point, fault_type] with aggregated statistics. Complexity scales as O(|D| \u0026times; |P| \u0026times; |F| \u0026times; circuit_cost). All circuit executions are independent and trivially parallelisable.\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eAlgorithm 2\u003c/strong\u003e \u003cp\u003e(Feature Attribution): For each qubit, sensitivity values are collected separately for inputs with (φ\u0026thinsp;=\u0026thinsp;1) and without (φ\u0026thinsp;=\u0026thinsp;0) the target feature. The attribution score is the difference in mean sensitivity. Statistical significance is assessed via t-test with Bonferroni correction.\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eAlgorithm 3\u003c/strong\u003e \u003cp\u003e(Causal Validation): For a qubit hypothesised to encode a feature, the algorithm computes prediction flip rates under fault injection for positive and negative examples. Validation succeeds when the positive flip rate exceeds 50% and is at least twice the negative flip rate. This dual criterion ensures the qubit genuinely encodes the feature rather than being generically fragile.\u003c/p\u003e \u003c/p\u003e"},{"header":"5. Experimental validation","content":"\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e5.1. Experimental setup\u003c/h2\u003e \u003cp\u003eWe evaluate FIP on three architectures: a variational quantum classifier (VQC) with 8\u0026ndash;12 qubits and 4\u0026ndash;6 layers, a quantum reservoir model (QRLM) with 12 qubits, and a quantum kernel classifier with 10 qubits. Datasets include SST-2 (1,000 sentiment examples), AG News (1,000 topic examples), and a synthetic dataset (500 examples) with known ground-truth feature\u0026ndash;qubit mappings. All circuits are simulated using PennyLane with 1,024 shots per execution. Results are averaged over five random seeds.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e5.2. Sentiment classification\u003c/h2\u003e \u003cp\u003eWe trained a VQC on SST-2 binary sentiment and applied FIP to identify sentiment-encoding qubits. Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e presents the bit-flip probing results.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eBit-flip probing results for sentiment classification on SST-2.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQubit\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSens. (Pos)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSens. (Neg)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eAttribution\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003ep-value\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQ3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.42\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e+\u0026thinsp;0.31\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.0001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQ7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.38\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.09\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e+\u0026thinsp;0.29\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.0003\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQ1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e+\u0026thinsp;0.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.847\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQ5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e+\u0026thinsp;0.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.612\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eQubits 3 and 7 show sensitivity three to four times higher for positive-sentiment inputs. Both p-values survive Bonferroni correction. Qubits 1 and 5 show nearly identical sensitivity regardless of sentiment, suggesting they carry non-sentiment information.\u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e presents the causal validation results through targeted interventions.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eValidation through targeted intervention on sentiment-encoding qubit candidates.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIntervention\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFlip Rate (Pos)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eFlip Rate (Neg)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eValidated\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBit flip Q3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e72%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e23%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eYes\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBit flip Q7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e68%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e19%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eYes\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBit flip Q1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e31%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e28%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePhase flip Q3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e45%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e18%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eYes\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eFlipping qubit 3 changes predictions for 72% of positive-sentiment inputs but only 23% of negative ones. This three-to-one ratio provides strong causal evidence. A phase flip on Q3 also validates (45% vs. 18%), suggesting sentiment is encoded in both computational basis and quantum phase.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e5.3. Synthetic ground truth\u003c/h2\u003e \u003cp\u003eTo confirm FIP does not detect artefacts, we constructed a synthetic dataset where feature φ directly affects qubits 2 and 5 during encoding. Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e presents the results.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eFIP attribution on synthetic dataset with known ground-truth mappings.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQubit\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTrue Encoding\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eFIP Attribution\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eCorrect\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQ0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eYes\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQ1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eYes\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQ2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eYes\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.34\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eYes\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQ3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eYes\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQ4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eYes\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQ5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eYes\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.41\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eYes\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQ6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eYes\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQ7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eYes\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eFIP achieves 100% identification accuracy. The lowest true-positive score (0.34) is nearly seven times the highest false-positive score (0.05). This clean separation confirms that the sentiment results are not spurious.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e5.4. Layer-wise information flow\u003c/h2\u003e \u003cp\u003eFIP can track information flow through successive circuit layers. Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e shows sensitivity evolution at four circuit points.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLayer-wise sensitivity evolution for sentiment-encoding qubits.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLayer\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eQ3 Sens.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eQ7 Sens.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eInterpretation\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePost-encoding\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eFeature not yet concentrated\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLayer 2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eFeature emerging\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLayer 4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.42\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.38\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eFeature concentrated\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePre-measurement\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.45\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.41\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eStable representation\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eWhat emerges is a concentration effect. Sentiment information starts distributed and progressively concentrates onto Q3 and Q7 through variational layers. By layer 4, the representation stabilises. This pattern mirrors classical transformers, where semantic features crystallise in middle-to-late layers.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec15\" class=\"Section2\"\u003e \u003ch2\u003e5.5. Comparing fault types\u003c/h2\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e compares the four fault models across our experimental suite.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eComparative analysis of fault types.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFault Type\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eAvg Sensitivity\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eBest for Detecting\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBit flip\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.34\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eClassical-like information\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePhase flip\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.22\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eCoherence-encoded features\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDepolarise (p\u0026thinsp;=\u0026thinsp;0.1)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eInformation robustness\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eErasure\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eInformation necessity\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eErasure faults produce the strongest signal by completely destroying qubit information. Bit flips serve as the best general-purpose probe. Phase flips uniquely detect quantum-native features encoded in coherence rather than population.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec16\" class=\"Section2\"\u003e \u003ch2\u003e5.6. Computational cost\u003c/h2\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e6\u003c/span\u003e reports wall-time measurements from our simulation environment.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab6\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eComputational cost scaling for FIP.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQubits\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eProbe Points\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTotal Circuits\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eWall Time\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e32\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e6,400\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e12 min\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e9,600\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e28 min\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e64\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e12,800\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e51 min\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e80\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e16,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e89 min\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eCost scales linearly with qubits and probe points. An 8-qubit analysis finishes in under 15 minutes. Circuit executions are independent, enabling 4\u0026times; speedup on a modest GPU cluster.\u003c/p\u003e \u003c/div\u003e"},{"header":"6. Applications","content":"\u003cp\u003eFIP supports four practical application areas. For model debugging, practitioners identify failing inputs, run FIP to locate qubits with anomalous sensitivity patterns, and diagnose root causes (encoding capacity, optimisation failures, or architecture limits). For architecture search, FIP characterises internal representations of competing designs; architectures concentrating features on fewer qubits with higher attribution tend to generalise better. For trust and deployment in high-stakes settings, FIP maps predictions to feature-encoding qubits, enabling auditability. For adversarial robustness, FIP identifies critical qubits with high sensitivity and low redundancy, estimating vulnerability to targeted quantum noise attacks without explicit attack construction.\u003c/p\u003e"},{"header":"7. Discussion","content":"\u003cdiv id=\"Sec19\" class=\"Section2\"\u003e \u003ch2\u003e7.1. Why FIP works\u003c/h2\u003e \u003cp\u003eFIP succeeds for four reasons. First, interventions establish causal links, not correlations. Second, quantum faults are natural operations\u0026mdash;we use the physics rather than fighting it. Third, we measure only final outputs, so collapse is not an obstacle. Fourth, differential comparison between feature-present and feature-absent inputs isolates specific contributions.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec20\" class=\"Section2\"\u003e \u003ch2\u003e7.2. Limitations\u003c/h2\u003e \u003cp\u003eFIP has notable limitations. Single-qubit probing may miss highly distributed, entangled encodings; multi-qubit coordinated probing addresses this at higher combinatorial cost. Hardware noise may mask injected faults on real devices; relative comparisons and increased shots mitigate this. Computational cost grows with dataset size, probe points, and fault types; adaptive sampling reduces overhead. Finally, FIP requires specifying features of interest rather than discovering them autonomously.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec21\" class=\"Section2\"\u003e \u003ch2\u003e7.3. Connection to causal inference\u003c/h2\u003e \u003cp\u003eFIP is grounded in Pearl's interventionist causal framework [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. Fault injections correspond to do-operations: P(Y|do(X_i\u0026thinsp;=\u0026thinsp;x')) rather than P(Y|X_i\u0026thinsp;=\u0026thinsp;x'). Interventions break confounding, ensuring FIP provides causal attribution rather than associative evidence.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec22\" class=\"Section2\"\u003e \u003ch2\u003e7.4. Extension to other architectures\u003c/h2\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab7\" class=\"InternalRef\"\u003e7\u003c/span\u003e summarises FIP adaptations across quantum model types.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab7\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 7\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eFIP adaptation across quantum model architectures.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eModel Type\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eAdaptation Required\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eVQC\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDirect application to variational layers\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQuantum Kernel\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eProbe the feature map circuit\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eReservoir Model\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eProbe reservoir and readout separately\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQAOA\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eProbe mixer and cost layers independently\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe reservoir model case is particularly informative. Probing the fixed reservoir reveals inductive bias, while probing the readout reveals what training extracted.\u003c/p\u003e \u003c/div\u003e"},{"header":"8. Conclusion","content":"\u003cp\u003eWe introduced Fault-Injection Probing, a framework for interpreting trained quantum ML models using controlled quantum errors as diagnostic probes. FIP is grounded in causal inference theory and respects quantum mechanical constraints. On sentiment classification, it identifies sentiment-encoding qubits validated through targeted interventions. On synthetic ground truth, it achieves perfect attribution accuracy.\u003c/p\u003e \u003cp\u003eFIP also enables layer-wise information flow tracking, fault-type comparison, model debugging, and adversarial robustness assessment. The framework is model-agnostic, extending to variational classifiers, quantum kernels, reservoir models, and QAOA. Limitations remain around distributed representations and computational scaling, but FIP provides a validated methodology where none existed before. As quantum models approach real-world deployment, the ability to interpret their learned representations will be essential for trust and reliability.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAuthor Contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003ePrabakaran Kannan: Conceptualization, Methodology, Software, Validation, Formal Analysis, Investigation, Data Curation, Writing – Original Draft, Visualization. Venkatesan M. Sundaram: Supervision, Writing – Review \u0026amp; Editing, Project Administration.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflict of Interest\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eEthics Statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis study is purely computational and involves no human participants, animal subjects, or personal data. All experiments were conducted using quantum circuit simulations on publicly available benchmark datasets. No ethical approval was required. Clinical trial number: not applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData Availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe code and experimental data will be made available on GitHub upon acceptance. All benchmark datasets used (SST-2, AG News) are publicly available.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eSchuld M, Petruccione F (2018) Supervised Learning with Quantum Computers. Springer, Cham.\u003c/li\u003e\n \u003cli\u003eLiu Y, Arunachalam S, Temme K (2021) A rigorous and robust quantum speed-up in supervised machine learning. Nature Physics 17:1013\u0026ndash;1017.\u003c/li\u003e\n \u003cli\u003eFujii K, Nakajima K (2017) Harnessing disordered-ensemble quantum dynamics for machine learning. Physical Review Applied 8:024030.\u003c/li\u003e\n \u003cli\u003eLipton ZC (2018) The mythos of model interpretability. Queue 16(3):31\u0026ndash;57.\u003c/li\u003e\n \u003cli\u003eWootters WK, Zurek WH (1982) A single quantum cannot be cloned. Nature 299:802\u0026ndash;803.\u003c/li\u003e\n \u003cli\u003eSimonyan K, Vedaldi A, Zisserman A (2014) Deep inside convolutional networks: visualising image classification models and saliency maps. In: ICLR Workshop.\u003c/li\u003e\n \u003cli\u003eSundararajan M, Taly A, Yan Q (2017) Axiomatic attribution for deep networks. In: Proceedings of ICML.\u003c/li\u003e\n \u003cli\u003eSelvaraju RR, Cogswell M, Das A, Vedantam R, Parikh D, Batra D (2017) Grad-CAM: visual explanations from deep networks via gradient-based localization. In: Proceedings of ICCV.\u003c/li\u003e\n \u003cli\u003eRibeiro MT, Singh S, Guestrin C (2016) Why should I trust you?: explaining the predictions of any classifier. In: Proceedings of KDD.\u003c/li\u003e\n \u003cli\u003eLundberg SM, Lee SI (2017) A unified approach to interpreting model predictions. In: Proceedings of NeurIPS.\u003c/li\u003e\n \u003cli\u003eBelinkov Y, Glass J (2019) Analysis methods in neural language processing: a survey. Transactions of the Association for Computational Linguistics 7:49\u0026ndash;72.\u003c/li\u003e\n \u003cli\u003eParis M, Řeh\u0026aacute;ček J (2004) Quantum State Estimation. Springer, Berlin.\u003c/li\u003e\n \u003cli\u003eNielsen MA, Chuang IL (2010) Quantum Computation and Quantum Information, 10th Anniversary Edition. Cambridge University Press, Cambridge.\u003c/li\u003e\n \u003cli\u003eSchuld M, Sweke R, Meyer JJ (2021) Effect of data encoding on the expressive power of variational quantum machine-learning models. Physical Review A 103:032430.\u003c/li\u003e\n \u003cli\u003eMcClean JR, Boixo S, Smelyanskiy VN, Babbush R, Neven H (2018) Barren plateaus in quantum neural network training landscapes. Nature Communications 9:4812.\u003c/li\u003e\n \u003cli\u003ePearl J (2009) Causality: Models, Reasoning, and Inference, 2nd edn. Cambridge University Press, Cambridge.\u0026nbsp;\u003cem\u003eWord count: approximately 3,200 words (excluding references and tables)\u003c/em\u003e\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Quantum machine learning, Interpretability, Explainable AI, Fault injection, Causal inference","lastPublishedDoi":"10.21203/rs.3.rs-8766996/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8766996/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eQuantum machine learning (QML) models achieve competitive performance on real-world tasks, yet interpreting what these models learn remains an open challenge. Classical interpretability techniques depend on access to intermediate representations, which quantum systems forbid due to measurement collapse, the no-cloning theorem, and exponential state-space dimensionality. We introduce Fault-Injection Probing (FIP), a framework that repurposes controlled quantum errors\u0026mdash;bit flips, phase flips, depolarising channels, and erasure\u0026mdash;as interpretability probes. FIP injects a known fault at a specific qubit and circuit layer, then measures the output shift. Comparing shifts across inputs with and without a target feature yields causal attribution scores linking qubits to learned representations. On variational quantum classifiers trained for sentiment analysis, FIP identifies sentiment-encoding qubits whose targeted perturbation flips 72% of relevant predictions. On a synthetic benchmark with known ground-truth mappings, FIP achieves 100% identification accuracy with zero false positives across all eight qubits. The framework is model-agnostic, extending to quantum kernels, reservoir models, and QAOA, and supports practical applications including model debugging and adversarial robustness assessment.\u003c/p\u003e","manuscriptTitle":"Fault-Injection Probing: A Causal Interpretability Framework for Quantum Machine Learning Models","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-02-09 13:58:25","doi":"10.21203/rs.3.rs-8766996/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"162a533a-73d7-4f6f-8784-485f6ea56b0a","owner":[],"postedDate":"February 9th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-02-09T13:58:25+00:00","versionOfRecord":[],"versionCreatedAt":"2026-02-09 13:58:25","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8766996","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8766996","identity":"rs-8766996","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}