arXiv CSopen access

Towards interpretable AI with quantum annealing feature selection

2026 · arxiv_cs

arXiv CS · Papers · License: Open Access · 2026

machine learning, deep learning, neural networks

Towards interpretable AI with quantum annealing feature selection Francesco Aldo Venturelli,1, 2 Emanuele Costa,2 Sikha O K,1 Bruno Juliá-Dı́az,3, 4 Miguel A. González Ballester,1, 2, 5 and Alba Cervera-Lierta2 1

arXiv:2604.25649v1 [cs.LG] 28 Apr 2026

BCN Medtech, Universitat Pompeu Fabra, Barcelona, Spain 2 Barcelona Supercomputing Center (BSC) 3 Departament de Fı́sica Quàntica i Astrofı́sica, Facultat de Fı́sica, Universitat de Barcelona, E-08028 Barcelona, Spain 4 Institut de Ciències del Cosmos, Universitat de Barcelona, ICCUB, Martı́ i Franquès 1, E-08028 Barcelona, Spain. 5 ICREA, Barcelona, Spain. Deep learning models are used in critical applications, in which mistakes can have serious consequences. Therefore, it is crucial to understand how and why models generate predictions. This understanding provides useful information to check whether the model is learning the right patterns, detect biases in the data, improve model design, and build systems that can be trusted. This work proposes a new method for interpreting Convolutional Neural Networks in image classification tasks. The approach works by selecting the most important feature maps that contribute to each prediction. To solve this combinatorial problem, we encode it into a quantum constrained optimization problem and propose to solve it using quantum annealing. We evaluate our method against the state-of-the-art explainable AI techniques, specifically GradCAM and GradCAM++, and observe an improved class disentanglement, i.e. the model’s decision boundaries become more distinct and its reasoning more transparent. This demonstrates that our approach enhances the quality of explanations, making it easier to understand which features the model relies on for specific predictions. In addition, we study the computational behavior of the quantum annealing algorithm. Specifically, we analyze the minimum energy gap of the system during computation and the probability that the algorithm finds the correct solution. These analyses provide theoretical insight into why the method works effectively in practice.

INTRODUCTION

Deep learning (DL) has emerged as the standard approach to solving computer vision problems, due to its ability to efficiently process large volumes of unstructured data such as images and videos [32, 49]. One of the most relevant contributions of DL to the development of the area of computer vision is attributed to the introduction of the Convolutional Neural Network (CNN) [26, 28]. CNN sequentially applies convolution operations that learn local spatial correlations within data in an efficient way, since the data-dimension is consecutively reduced across multiple layers while higher-level features are extracted. Different CNN architectures have been proposed and explored so far. The first breakthrough in CNNs began with AlexNet, which achieved significant performance on the ImageNet [44] challenge and sparked a wave of architectural innovations [26]. Subsequent architectures such as VGGNet [47] and ResNet [20] demonstrated that deeper structures and residual connections could significantly improve representation learning and training stability, becoming widely used backbone models for visual recognition systems [24, 38]. Subsequently, many other variants of CNNs have been proposed for various pixel-level tasks such as U-Net [43] that introduced an encoder–decoder scheme with skip connections and became a standard approach for image segmentation [2, 39]. Modern CNNs are also used in generative modeling, including variational autoencoders and generative adversarial networks [18, 19, 25], diffusion models [10, 22, 41, 48] and vision transformers [15, 50].

Despite their success, DL models often behave as “black boxes”. Their predictions emerge from complex sequences of operations involving millions of trainable parameters, rendering internal decision processes difficult to interpret [6, 36, 54]. This lack of transparency poses a significant challenge in safety-critical applications where understanding model behavior is essential for establishing trust and accountability [14]. To address this limitation, the field of explainable AI (XAI) has emerged, offering a suite of techniques designed to inspect model outcomes and highlighting the reasoning behind specific predictions [45, 52]. XAI methods are broadly categorized into ante-hoc, which incorporate interpretability constraints during model design, and post-hoc approaches, which analyze trained models after the fact. In the field of interpretability, CNNs are relatively easy to work with since a wide set of studies have been conducted and post-hoc techniques are commonly applied to such architectures. Among them, gradient-based activation mapping methods have gained considerable attention. Examples are Grad-CAM [46] and GradCAM++[7], which compute the gradients of a target class score with respect to the final convolutional layer, producing coarse localization maps that highlight image regions that most influence the prediction. These methods, however, operate at an aggregate level. They combine contributions from all feature maps (FMs) into a single saliency visualization without disentangling the role of individual FMs within the network. This aggregation obscures which specific learned representation is responsible for a given prediction. A promising direction to address the limitations of

2 these gradient-based mapping methods lies in feature selection (FS), a well-established paradigm that has gained renewed relevance in the context of XAI. Traditionally, FS has been predominantly employed as a data preprocessing step, used to reduce input dimensionality, remove noise, and improve training efficiency before model learning begins [23]. This conventional view has led to FS being underutilized as a tool for model interpretation. However, when applied to learned representations rather than raw input data, FS offers a powerful mechanism for post-hoc explainability that remains largely underexplored. Some recent works have recognized the potential of FS principles within the XAI paradigm [8, 29, 40, 50, 53]. However, it is known that FS is computationally challenging, since searching a subset of features that satisfies a certain condition scales exponentially with the number of features in the search space. For this reason, several heuristic and meta-heuristic approaches have been proposed to address this problem. In this work, we explore the novel use of quantum computing for FS, in the context of XAI. Quantum computing provides a natural framework for encoding combinatorial optimization problems, such as FS. In particular, by mapping the FS task into a Quadratic Unconstrained Binary Optimization (QUBO) formulation, each binary variable can represent the inclusion or exclusion of a feature, and the objective function can be designed to capture relevance, redundancy, and task-specific constraints. Within this framework, quantum computers enable the efficient exploration of the exponentially large solution space. This process can be interpreted as a guided search over the space of feature subsets, where multiple candidate solutions are implicitly considered in parallel and progressively driven toward optimal or near-optimal configurations. Reformulating FS schemes as a QUBO problem is not a novel attempt, yet the majority of the proposed approaches primarily operate on input-level features rather than the internal learned representations of deep networks [4, 16, 21, 31, 42]. Although current Noisy Intermediate-Scale Quantum (NISQ) devices [5] do not yet guarantee a quantum advantage, they offer a promising paradigm for tackling high-dimensional FS problems, especially when combined with classical preprocessing or hybrid optimization strategies. In this work, we propose a quantum annealing (QA)based FS approach for generating the bit-strings containing the selected FMs: the explanation maps of CNNs. By formulating FS as a QUBO problem, we leverage QA to efficiently identify the subset of FMs that most strongly contribute to a given prediction. The novelty we introduce relies in the fact that we do not apply FS directly on input pixels. Rather, we consider the hidden representations that the model learns at a particular stage as the elements over which performing the selection. Practically speaking, we enforce the QA to sample the bit-string that maximizes the linear term representing the importance of a single FM to the prediction, while minimizing the ge-

ometrical similarity among distinct FMs through the cosine similarity term, in a way similar to the well-known NP hard Maximum Diversity problem [17, 27, 30]. In this way, we combine the core aspect of GradCAM (gradient importance) with a proper quadratic term (that considers the mutual geometrical similarity) to construct the QUBO problem. Reducing the number of FMs by FS schemes provides a natural strategy to mitigate scalability and dimensionality challenges in DL models. Additionally, it improves model interpretation by isolating FMs that strongly contribute to a given prediction, compressing and discarding similar and redundant representations. In this way we characterize the most informative patterns that the model learns during the training process to be able to interpret the resulting predictions. The article is organized as follows: in sec. II we give a detailed description of the FS algorithm, defining the mathematical formulations and the QA simulation procedure. Subsequently, we present the main results, i.e. the explanation maps sampled by QA and the class-class correlation maps that quantify the disentanglement among distinct image classes. Then, we analyze the model’s success by defining physics-derived metrics, such as the minimum energy gap between the first excited state and the ground state of the QUBO Hamiltonian and we explain the meaning behind these results. In the sec. III, we discuss the relevance of our approach by explaining the key ideas behind the application of QA to the framework of interpretability, while highlighting our contributions, the novelty with respect to the state of the art, and the possible future directions that can be further explored.

II. A.

RESULTS

Quantum annealing feature map selection algorithm

As mentioned in the introduction, we propose a XAI method that identifies the relevant FMs needed to classify a given image. To do so, we encode the optimization problem of finding this set into a QUBO problem and solve it with QA. The overview of our algorithm, as shown in Fig. 1, is the following: we take the FMs produced from the last convolutional block and select those whose gradient positively contributes to the model’s output (importance measure). We then generate an Isingtype Hamiltonian using the absolute value of the cosine similarity between the FMs and the importance of each FM. The QA evolves the corresponding Hamiltonian to find the ground state containing the superposition of the FMs relevant for the image. By repeating this process for the whole image dataset, we extract a class correlation map that allows us to interpret which classes share common FMs. In other words, we disentangle the FMs from the CNN to obtain only those relevant to identify the image class.

FIG. 1. Representation of the FS algorithm reformulated as a QUBO problem. 1 − 2) A single image is fed into a CNN that has (a) been previously trained. From the last convolutional layer we extract the FMs fij (x), with (a) ∈ Nf . 3) A one-hot encoding is applied to the FMs that contribute positively to the gradient. 4) We construct the global problem Hamiltonian, prepare the initial quantum state in superposition of all possible Nf combinations and we use QA to evolve the state towards the minimum of the QUBO Hamiltonian. 5) Finally, we sample the resulting bitstring, which corresponds to the explanation map containing the FMs that satisfy the constraints imposed by the problem Hamiltonian: maximizing the positive contribution of each FM while minimizing their geometrical similarity. This pipeline is repeated for M images from the test set.

Formally, a CNN is a parametric function composed of convolutional operators that map an image x ∈ RH×W ×C , where H and W are the height and width of the image (e.g. the number of pixels that it contains), and C is the channel (e.g. C = 3 for RGB), to hierarchical FMs, followed by a dense classifier operating on the learned representations. Each convolutional layer takes as input the FMs produced by the previous layer and applies learnable filters to extract higher-level features; the spatial dimensionality is reduced through sequences of striding or pooling operations. In a ResNet architecture, each convolutional layer produces a fixed number of FMs determined by the number of trainable filters. While the number and dimensions of these FMs are the same for all inputs, the activation values within them differ for each image because they depend on the specific patterns present in the input. Therefore, after training the CNN on a given dataset, each input image produces a set of FMs whose activation values depend on the patterns present in that image. While images belonging to the same class often generate similar activation patterns, the learned filters themselves are not explicitly labeled, and it is generally not straightforward to interpret what specific semantic feature each filter represents. Given the set of FMs from the last convolutional layer,

{f } ∈ RHf ×Wf ×Nf , where Hf and Wf are the height and width of the last layer (typically, way smaller than the image H and W ) and Nf is the number of the learned representations (that depends on the CNN architecture, but typically much larger than C), and the final output of the dense classifier, z = CNN(x), where CNN represents the full CNN processing layers, we compute the following quantity αa =

Hf ,Wf X ∂z 1 , Hf Wf ij ∂f (a) ij

(1)

for each FM a, which is the global average pooling of the gradient. This quantity is equivalent to obtaining a scalar measure of the importance for each FM. In the GradCAM approach, αa is just the weight of the corresponding FM. Depending on the sign of αa , we can distinguish class-confirming FMs (positive sign), which contribute to characterizing the class, and class-opposing ones (negative sign) that pivot on characterizations that lower the confidence of the classification. In our approach, we propose a filtered GradCAM, i.e. f GradCAM, by selecting the learned representations that satisfy {f̃ } ≡ {f (a) |αa > 0}.

(2)

4 Notice that each f̃ ∈ RHf ×Wf and there will be d ≤ Nf filtered FM. Next, we compute the cosine similarity between the pairs of filtered FM, P (p) (q) | ij f˜ij f˜ij | (p) (q) . (3) Jpq = |⟨f̃ ; f̃ ⟩| = ||f̃ (p) || ||f̃ (q) || This quantity is large when there is overlap and, therefore, redundancy between distinct FMs, while it is small when the FMs are more geometrically independent. The idea behind the cosine similarity is to assume that FMs that encode similar information, which is redundant for the model, are parallel vectors in the feature latent space. Thus, we look for the subset of vectors that are maximally orthogonal to each other. Our goal is to select the subset of positive representations maximizing the importance of the filtered FM, α̃ > 0 (gradient) and minimizing the redundancy Jpq (cosine similarity) between the filtered FMs. To do so, we propose a one-hot encoding for the filtered FM, rep1−σz(p) resenting each of them as a unary vector n̂p ≡ , 2 where σz = diag(1, −1) is the Pauli-Z operation acting on qubit p. Therefore, we will need d qubits to represent the space of filtered FMs. Next, we construct a QUBO Hamiltonian of the following form: X 1X ĤQUBO = (1 − β) Jpq n̂p n̂q + β hp n̂p . (4) 2 pq p The transverse field encodes the importance of the filtered FMs α̃p hp = , (5) max α̃p such that its values range in hp ∈ [0, 1]. Finally, β is a hyperparameter that defines the interplay between the two terms. For β = 1 all the positive FMs are selected. Conversely, smaller β values weaken this bias and encourage QA to explore a broader subset of FMs, thereby incorporating contributions from less dominant gradient components. In the limit β = 0, only cosine similarity is considered, i.e. there are no constraints on the number of selected FMs and the solution will be the trivial null set. In our analysis, we set β = 0.7 as we numerically observed the results were better if we gave slightly more weight to the importance. The QA protocol starts with initializing the qubits in the ground state of a driver Hamiltonian ĤD = Pd (p) − p σ̂x , i.e. in the full superposition state of all the bit-strings that encode all possible subsets of FMs Nd P 1 |ψ⟩init = p=1 |+⟩p , i.e. |ψ⟩init = 2d/2 z∈{0,1}d |z⟩. Here, 0 1 σx = represents the Pauli−X operator acting on 1 0 qubit p. The dynamics is governed by a time-dependent Hamiltonian Ĥ(s) = A(s)ĤD + B(s)ĤQUBO ,

(6)

where s is the dimensionless time parameterized as s ≡ τt , where τ is the total evolution time, t is the time and A(s) and B(s) satisfy the conditions B(0) = 0, A(1) = 0. In our case, we chose a linear interpolation between A and B, i.e. A(s) = 1 − s and B(s) = s. Notice that the combinatorial optimization problem grows exponentially with the number of filtered FM d (the Hilbert space has dimensions 2d ), while by solving it with a QA protocol, the final solution after the evolution is sampled using a polynomial number of shots, typically nshots = O(d2 ).

Benchmark

To validate the proposed method, we take the STL-10 [9] dataset that contains 5000 images in total that belong to 10 distinct classes (for more details, check the IV A section). As a CNN architecture, we use a pretrained ResNet-18. An important point in our benchmarks is that we need to constrain the maximum number of FMs in the last convolutional layer, Nf , to simulate the QA protocol. If we had access QA hardware, this step would not be necessary, since state-of-the-art QAs contain more than 1000 qubits. We study the scalability of our simulations by increasing Nf from 16 up to 32 FMs. Moreover, we also run a simulated annealing (SA) protocol with all Nf = 512 FMs to obtain an estimation of the protocol’s success, although this final simulation will not give the relevant physical properties necessary to understand the physical limitations of this protocol. We start by obtaining the confusion matrix by computing the Bhattacharyya coefficient [35], which estimates how likely two classes share the same FMs. A fully diagonal correlation matrix indicates well-separated (i.e., disentangled) FMs that capture information specific to a single image class. In contrast, non-zero off-diagonal elements reveal shared representations among different image classes and constitute the primary source of falsepositive and true-negative errors. Figure 2 shows the class-class correlation map for a CNN restricted to Nf = 16 filtered FMs (top) and for the full ResNet-18 with all the Nf = 512 FMs. In the first case, we can analyze the physical aspects of the protocol, as will be shown in later sections, while for the full ResNet case we use SA. In both cases, the matrix shows an overall good disentanglement across different image classes. Focusing on the Nf = 16 case, we obtain only 15 non-zero (≥ 0.1) overlap out of the 44 possible image class combinations, and all of these lead to a Bhattacharya coefficient below 0.5. We leave a formal explanation of the results in the section III. For the full FMs simulation, we mostly disentangle completely all the classes. We then test our proposed FS against well-known classical methods that have achieved promising results and constitute the state-of-the-art in the context of interpretability. Specifically, we compare the explanation

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FIG. 2. (Top) Class–class correlation map for the FS algorithm for a ResNet-18 constrained to Nf = 16 FMs in the final layer and trained for 20 epochs. Warmer colors indicate stronger overlap between FMs. The matrix is obtained by computing the scalar product between the distributions of the activated FMs per image class, i.e. the Bhattacharya coefficient. In the QA simulation, we set τ = 50 and nshots = d2 , where d is the number of FMs that contribute positively to the output’s gradient. A full diagonal matrix indicates complete orthogonality among classes. Values below 0.10 are rounded to 0. (Bottom) Class–class correlation map for FS algorithm using SA for the full ResNet-18 model trained for 100 epochs. For Nf = 512, the SA is able to find a good disentanglement between hidden learned representations. Quantitative analysis - Average Drop % Nf = 16 Nf = 512 Class GradCAM GradCAM++ QA-f GradCAM GradCAM GradCAM++ SA-f GradCAM 0 Airplane 33.1 17.1 13.8 1.20 2.29 12.9 1 Bird 14.4 13.4 7.06 4.76 4.02 23.3 2 Car 3.87 0.44 0.67 0.97 1.38 8.09 3 Cat 11.6 17.0 4.99 4.29 1.29 14.0 4 Deer 24.8 4.21 14.5 5.11 5.27 6.47 5 Dog 63.3 56.65 56.9 12.1 17.0 30.1 6 Horse 7.90 8.75 3.43 7.31 12.4 32.6 7 Monkey 16.2 6.36 4.18 13.0 18.9 34.0 8 Ship 0.72 0.25 0.70 0.97 1.59 14.9 9 Truck 0.22 1.11 0.14 2.88 3.01 7.37 x ± σx 17.6 ± 5.74 13.2 ± 4.97 10.6 ± 5.11 5.26 ± 1.31 6.73 ± 2.04 18.4 ± 3.22 TABLE I. Comparison between GradCAM, GradCAM++ and our QA-f GradCAM protocol in terms of the Average Drop % per image class. The QA simulation takes τ = 50 and nshots = d2 , where d is the number of filtered FMs for each image. Out method performs better than GradCAM and in most classes also better than GradCAM++.

maps obtained by our protocol with Nf = 16 against the spatial projections originated by GradCAM and GradCAM++ for the entire set of image classes. As a qualitative example, Figure 3 shows different explanation maps that highlight the images for the specific class of airplane images. The explanation map derived by the QAf GradCAM is a bit string containing the FMs that satisfy the constraints of the QUBO problem (positive gradient contribution and maximal orthogonality between

FMs), as explained in the previous section. In this way, we expect to obtain GradCAM and GradCAM++ which contain less spurious representations compared to the filtered one. To obtain a quantitative analysis, we estimate the Average Drop % [7] (see Sec.IV D for details) which essentially replaces all the pixels of the image that are excluded by the explanation map to 0 (i.e. set them to black), and then inputs this new image to the model to

6 Class: Airplane

QA-filtered GradCAM

GradCAM

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QA-filtered mask

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FIG. 3. Qualitative comparison example of GradCAM and GradCAM++ with our QA-f GradCAM protocol for Nf = 16 and τ = 50. The selected representations capture relevant regions and seem to identify parts of the images that are misinterpreted by the state-of-the-art methods.

Model’s efficiency and success

We now proceed to analyze in detail the physical properties of the QA protocol to address its efficiency and success. In QA, a relevant quantity to determine the model’s complexity is the energy gap ∆min ≡ E1 − E0 between the ground state and 1st excited state energies of the time-dependent Hamiltonian Ĥ(s) from Eq. (6). If ∆min tends to zero, the annealing time parameterized by s must be longer to ensure that the system remains in its ground state (for more details, see Sec. IV F). We start with an aggregated analysis of the gap for all the dataset. For each image, we diagonalize Ĥ(s) to obtain its spectrum and compute the ∆min for time steps of ∆s = 0.01 and τ = 10. In Fig. 4 we plot the histogram and the cumulative curve. We observe that the overall trend is placed towards large energy gap values, near 10−1 unit of energy, while the probability of having val-

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calculate the prediction score (accuracy). Therefore, the lesser Average Drop %, the better, which implies that the explanation map obtained was already optimal and that change in the pixels does not significantly modify the accuracy of the model. Table I shows the Average Drop % per class for the different methods. QAf GradCAM achieves on average better results compared to GradCAM and GradCAM++. From the 10 classes, only in 4 of them GradCAM++ achieves better results than QA-f GradCAM. On the contrary, when the initial number of FMs is increased to 512, our method shows worse behavior compared to the classical counterparts. Despite the apparent challenge, we give in the next section III a possible explanation on why this potentially occurs for the current set-up of parameters we use.

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ues below 10−3 is very close to 0. This indicates that the QA evolution is sufficient and robust to converge to the ground state of the target problem Hamiltonian ĤQUBO for most of the instances of the QUBO problem. Next, we investigate the probability of success, i.e. of finding the true ground sate, of the QA protocol for each image. We compute the fidelity F (i.e. a measure of distance) between the quantum state obtained after the QA evolution and the exact ground state of Ĥ(s). For a random final state, F̄ ≃ 21d , i.e. a uniform distribution. In general, any scaling of F of the type F ∝ d−γ , with γ > 0 is considered a good result, since it restricts the research method to a subspace polynomial with the degrees of freedom of the system. One would expect that for low values of τ , the evolu-

III.

DISCUSSION

DL architectures such as ResNet-18 can produce hundreds of FMs at a single convolutional layer. Identifying the most informative and non-redundant subset for a given prediction requires searching through a large solution space that scales ∼ 2Nf , and classical FS algorithms may struggle to scale efficiently under such conditions. This fact motivates the use of quantum computers, in particular QA, machines specially designed to solve combinatorial optimization problems by sampling from the problem solution obtained after the quantum evolution. Our work introduces a novel approach to assess interpretability of large models, such as CNNs, by characterizing the learned FMs extracted from a target convolutional layer, and efficiently selecting those that contribute to uproot class-specific information from the data samples. Our approach is motivated by the desire of performing a characterization of the entire set of FMs that compose the image. Each of them is appropriately inspected, revealing for which image class it is activated and whether it is shared among distinct images. We also achieve a significant model simplification, incrementing the level of confidence in interpreting predictive outcomes, by directly modeling the learned set of representations instead of focusing on the architecture itself. To do so, we apply a FS algorithm to perform FMs characterization. Subsequently, we reformulate the FS algorithm as an energy minimization problem to be solved by a QA. As shown in Fig. 2, while obtaining a quite disentan-

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tion will not be adiabatic and the system will end up in an excited state instead of the ground state. Therefore, this measure will also illustrate what is the scale of the annealing time τ necessary to archive accurate results. Figure 5 (left) shows the median value F for each class of the dataset and for different values of τ . Since the number of filtered FMs d may vary from image to image and that measure is what determines the number of qubits of the QA protocol, we also compute F as a function of d and show it in the right part of Fig.5 (right). In both figures, the error bars represent the 1st and 3rd quartile of the samples. For τ = 10 the system falls into a diabatic regime, i.e. too fast, and the result does not correspond to the ground state. In general, we observe that for τ > 50 the solution is obtained with higher probability. Also, there is no significant difference between the behavior as a function of the class or as a function of the number of filtered FMs d. Therefore, although the dimension of the Hilbert space increases exponentially for d, we still obtain higher fidelity values overall, meaning the QA is able to end up into the ground state of ĤQUBO for the majority of the problem instances. Finally, we explore the diabatic-adiabatic behaviour for higher number of FMs Nf in the last convolutional block, which allow us to obtain the Landau-Zener dependence, as expected (Fig.5).

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FIG. 5. (Top) Median fidelity F values as a function of the class for different values of τ and Nf = 16. For lower values of τ it exhibits a scaling behavior approximately proportional to ∼ d1 , indicating increased difficulty as the problem size grows, as expected. Given that the distribution of the minimum gap is shifted towards larger values, as τ increases, we notice a suppression of this scaling as the evolution achieves an adiabatic regime. (Bottom) Median F values as function of the filtered FMs d for distinct τ with different Nf = 16, 24, 32. For higher values of τ , the fidelity shows a global constant trend, while for τ = 10, in diabatic regime, we observe an exponentially decreasing trend, in accordance with the Landau-Zener formula from Eq. (8).

gled correlation matrix, our model also learns representations that are commonly found in distinct image classes. For instance, classes representing animals may show a non-zero overlap since many of them are characterized by the same number of legs, similar color patterns or similar overall shape. Similarly, airplane images may be confused as ships or trucks because the overall shape of singular structures present in both classes is similar. In addition, we are assuming all the FMs that are selected belong to the subject of the classification. Rather, it may happen that a positive contribution to the gradient is obtained from a FM that is part of the background, and its selection is enforced through the quadratic term (it may be a geometrically independent vector with respect to those that contain information of the image subject). In this case, we can have distinct background-related FMs that are shared and activated within distinct image classes. Nevertheless, the feature characterization serves at this scope: understanding which filters are activated

8 among distinct classes to further drive a more sophisticated training process that would take this into account. To benchmark the interpretability of the selected FMs, we evaluate the Average Drop % of the resulting CAMs. As shown in Tab. I, the quantum FS with Nf = 16 features achieves a lower Average Drop than full GradCAM and GradCAM++, confirming that QUBO-driven redundancy reduction yields more discriminative feature representations. Nevertheless, the Average Drop % is sensitive to the number of selected FMs and thus to the interplay parameter β in the QUBO formulation from Eq. (4), as it is observed in the results obtained in the Nf = 512 model, also presented in Tab. I. For the targeted convolutional block of the full ResNet18, SA fails to recover a FM subset that faithfully represents the GradCAM and GradCAM++ class activation maps. Two compounding factors explain this result. First, verifying whether the SA solution reaches the global minimum of the QUBO problem is infeasible at Nf = 512, since neither simulated QA nor brute-force search scale to this problem size. Second, the spatial resolution of the FMs extracted from the last convolutional block of the full ResNet-18 is lower than in the custom architecture (Nf = 16), leading to larger pairwise cosine similarities Jpq and a more aggressive selection fore the same β. As a result, the quadratic redundancy term in ĤQUBO suppresses a larger fraction of positively weighted FMs, leaving only a small surviving subset. For a fixed β, this produces explanation maps with degraded Average Drop %. This case study underscores the importance of tuning β until the Average Drop % converges to that of the reference CAMs. Furthermore, introducing a cardinality penalty constraining the number of selected FMs while leaving the redundancy term unaffected could improve the resulting explanation maps, at the cost of a bias toward a fixed number of selected features. In terms of model’s complexity, we analyze in IV the couplings Jpq . The distribution is well approximated by a Gaussian, except for an initial bin peaked in correspondence of 0 value of Jpq that coincides with the main diagonal of the cosine similarity matrix, which is 0 accordingly. The fact that the distribution is unstructured and represents a Gaussian means that our Ising model from the QUBO formulation falls into the well-known SherringtonKirkpatrick model [11, 33], which, in its hardest instance is NP-Hard [3]. This result is consistent with the wellknown averaging effect in deep networks: as the number of FMs grows, individual feature representations become increasingly correlated and their pairwise overlaps selfaverage, suppressing large fluctuations. As a result, the effective QUBO instance is far from the hardest SK configurations, yielding a tractable optimization landscape. Tuning β parameter may increase the complexity of the problem if combined with a more structured Jpq distribution. In fact, other datasets can present more sophisticated coupling distributions that could not be estimated in a classical way. The choice of not including a constraining term that tunes how many FMs are part of

the final solution bit-string simplifies the overall problem formulation, but this can be easily extended to include such a penalty term. As we can see from Table I, applying the proposed methodology to earlier convolutional blocks, where FMs do not yet exhibit a clearly interpretable semantic structure, can still improve the activation of those FMs that significantly contribute to the final prediction. A possible explanation is that performing FS at these early stages effectively acts as a structured information filtering mechanism. By suppressing redundant FMs before higher-level representations are formed, the CNN is encouraged to propagate only the most relevant low-level features. In this sense, the selection process can be interpreted as a form of information transfer: informative components that might otherwise be discarded by subsequent layers are preserved and transferred throughout the forward pass. This effect becomes particularly evident in the deeper convolutional blocks and in the classifier, where the representations appear more focused and discriminative. Therefore, early-stage FM selection may improve the overall efficiency of representation learning by reducing redundancy and reinforcing task-relevant features across the network hierarchy. Although our approach is model-dependent, meaning results can vary with different model structures or training setups, it is flexible and can be extended to a wide range of alternative unsupervised and generative learning models. As explained above, our work is universal: the FMs selection procedure can be sequentially applied to any intermediate convolutional block to study which trainable filters are activated across the whole network. We can also use our approach to reverse-engineer a FM search by penalizing the selection of orthogonal FMs and thus restricting our model to find those that are maximally aggregated. In conclusion, our novel approach to address the FS problem in DL and the QA mechanism that can deal with large optimization landscapes is flexible enough to be extended as an agnostic tool for explainable AI.

IV. A.

METHODS Dataset

We use the STL-10 dataset [9], which is a dataset used primarily for image classification. It consists of 10 distinct classes (airplane, bird, car, cat, deer, dog, horse, monkey, ship, truck) of 96 × 96 pixels RGB images, with 4750 training and 250 test samples. We extract 250 images from the training set to construct the validation set. All the images are acquired from labeled examples from ImageNet. We upsample the original images to a dimension of 224 × 224, since this corresponds to the dimension of the data samples within ImageNet that are used to train the ResNet-18.

9 B.

CNN architecture

As CNN architecture, we use a pretrained ResNet-18 [20], originally trained on the ImageNet dataset [13]. For transfer learning, we replace the last convolutional block and the classification tail with custom modules, ensuring a reduced number of final class-specific features, i.e. from 256 to Nf . Specifically, we build such layer preserving the structure of the original ResNet layer, including BatchNorm and ReLU activations after two Conv operations that take as input 256 FMs and output 256 in the first, and Nf in the second. We also introduce ReLU activation functions at the end of each layer. While our algorithm is compatible with the full ResNet-18, reducing the number of final FMs makes the analysis of the QUBO Hamiltonian computationally feasible. In the first attempt, we consider the model with a final convolutional block producing Nf = 16 FMs. Subsequent scalability experiments involve models with Nf = 24, 32 FMs. For the SA simulation we consider the full ResNet-18 architecture with all FMs (Nf = 512). This architecture includes a Global Average Pooling layer at the end that compresses the entire information to Nf = 512 ∗ 1 ∗ 1.

General Deep Learning setup

Transfer learning is performed using the PyTorch library [34]. We adopt the Adam optimizer [25] to finetune the parameters of the additional convolutional block and the classification tail. The convolutional block consists of a sequence of 256 input channels and Nf = 16 (in the scalability experiments, we increase it up to 32) output channels, followed by batch normalization and a ReLU activation function. The classification tail is a dense layer of dimension Hf × Wf × Nf , where Hf × Wf is the spatial size of the FMs. We trained the network for a limited number of epochs Nt = 20, and we store the value of the validation accuracy after each epoch. We then consider the model configuration that has reached the highest value of validation accuracy to avoid overfitting, and we finally perform the evaluation on the test set, obtaining a test accuracy of acc = 94%. The training phase influences the final results we obtain: a longer training helps the model identifying additional and finer FMs. We consider that Nt = 20 epochs with that accuracy is enough to give meaningful results and that increasing the training time will not modify the conclusions substantially. After model training, the most important part at this stage is the computation of the output gradient with respect to the single FM extracted by the convolutional wedge, which is done by the PyTorch built-in hook method. It is a powerful mechanism for gaining insights into the behavior of the CNN during both forward and backward passes. We use the forward hook that allows us to retain the output of the model after a single forward pass. By hooking the model, we are able to

Train Nf = 16 Nf = 24 Nf = 32 Nf = 512

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FIG. 6. Example of the checkpoints. The train and validation losses are evaluated at each 5 epoch from a total of Nt epochs and stored into corresponding model’s checkpoints for Nf = 16, 24, 32 and 512. L corresponds to the cross-entropy loss function.

compute the gradient of the model’s output with respect to the activation. After the modified gradient computation, we eliminate the corresponding hook to move to a subsequent data point. The last important detail to consider is that we use two different versions of the same ResNet-18 model, as mentioned in IV B. In the first version we consider the FMs extracted from the custom convolutional layer inserted after the third layer of the original ResNet-18, while for the second version we get the full set of FMs from the fourth original layer, before the classifier. In the first version, the FMs belong to the third layer and the information is compressed to originate a number Nf of these ones. Thus, their resolution is different with respect to those that belong to the original full ResNet-18. D.

Average drop

In our benchmark analysis we compute the Average Drop % [7], which calculates the percentage of drop in the model’s confidence for a particular image class when is represented by its explanation map. It is formally defined as M

1 X max(0, Y c (xi ) − E c (xi )) ·100, M i Y c (xi ) (7) where, c corresponds to a particular class, Y c (xi ) refers to the model’s output score (confidence) on the image xi , and E c (xi ) is the same model’s confidence with only the explanation map region as input. If the Average Drop computed on the selected FM subset exceeds that of the full GradCAM, the subset fails to capture the discriminative information carried by the complete set. Conversely, a lower Average Drop indicates that removing redundant FMs improves representability, validating the selection. Average Drop % =

Quantum annealing simulation

The QA simulation is done using matrix and tensor product multiplication to build the global time-

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FIG. 8. Gaussian fit of the Jpq distribution across all dataset. The distribution shows a Gaussian shape. The peaked bin corresponding to 0 value of Jpq represents the diagonal term of the cosine similarity matrix.

passing through an avoided crossing is given by FIG. 7. (Top) average Drop % for SA with different β. (Down) bit-string length as a function of β, divided by Nf = 512. As visible, by increasing the value of β the linear term becomes more important within the QUBO formulation, and we converge to the results we would obtain with standard GradCAM and GradCAM++.

dependent Hamiltoninan Ĥ(s) from Eq. (6). From the n̂ operators, the matrix exponentiation built-in method from the Scipy [51] library is used to perform the quantum state evolution in the same fashion of the SuzukiTrotter formula. To analyze ∆min and compute the fidelity F , we apply the Lanczos diagonalization method [12] to extract the eigenvalues and eigenstates of Ĥ(s) at τ each time step s = 0.01 .

F = 1 − ProbLZ = 1 − e−λ∆ ,

(8)

where λ is some constant. As the energy gap ∆min reduces, we expect a maximum ProbLZ value, meaning that the system likely jumps to higher energy levels rather than following the adiabatic theorem. While for larger τ values, we remain in the adiabatic condition. Finally, it is important to make a further consideration regarding the couplings Jpq and the hyperparameter β. As visible from Fig. 8, the couplings distribution is well approximated by a Gaussian, meaning that our Ising model falls into the well-known Sherrington-Kirkpatrick model [11, 33]. This implies that this particular Ĥ(s) can have a mean field approximation, i.e., be solved classically.

Complexity of the QA protocol G.

Following the approximate adiabatic theorem [37, 55], the time evolution τ needed to end in the ground state at the end of the protocol, scales as τ ∝ ∆−2 min . Therefore, one needs to prove that ∆min does not decrease exponentially with the system size to conclude that QA can be used to efficiently solve a QUBO problem of the form of Eq. (4). Otherwise, one would need exponentially long τ to remain in the ground state. Regarding the analysis of the values of τ necessary to achieve good fidelities, we notice in Fig. 5 that for lower annealing time values τ ∼ 10, the model falls into the diabatic regime. In such condition, the probability of success at the end of the annealing process follows a negative exponential scaling as depicted in the red dashed line. This is in agreement with the Landau-Zener formula [1]. In fact, in diabatic regime the probability of

Simulated Annealing

We build a modified version of the full ResNet-18 model where we attach ReLU activation layers after each convolutional block, as described in IV B. We then extract the entire set of FMs composed of 512 elements and we apply the QUBO formulation to obtain the bitstring corresponding to the problem solution. The linear term of the QUBO matrix is composed with the positive gradient contribution of the d FMs, while the off-diagonal terms contain the relative cosine similarity between them. We set a number of num reads= 1000, num sweeps= 1000, num sweeps per beta= 1, beta schedule type= linear as default, and each read is generated by one run of the SA algorithm, as reported in the documentation of Dwave Ocean, which is used to build and execute SA.

101

CODE AND DATA AVAILABILITY

fit: median d 1.24

Code link https://github.com/checc1/FS_QA min

10 1

ACKNOWLEDGEMENTS

V. A.

SUPPLEMENTARY MATERIAL

More analysis on the complexity in the Quantum Annealing protocol

The complexity of a QA protocol depends on the energy gap ∆min . We explore the scaling properties of ∆min to verify if the QA protocol is feasible on machines with limited τ and to quantify the advantage of the method. As shown in Figure 9, the average ∆min for different filtered FMs d scales O(d−1 ), which proves that the scaling of τ is quadratic with the system size. We also plot the hardest instances, ∆∗ , i.e. the cases with the smallest energy gap to show how the it distributes across distinct dimensions d. Another important aspect is to identify the time tmin ≡ t(∆min )/τ when the evolution finds the minimum energy gap. This information will help in identifying more efficient annealing protocols, i.e. in defining the A(s) and B(s) functions from Eq. (6). In Figure 10 we plot the distribution of tmin /τ as a function of ∆min for all the problem instances of different image classes. We also depict the distribution of the values of time tmin , indicating that for the linear scaling of the two Hamiltonian terms, the minimum gap is obtained at the end of the annealing process.

10 2 10 3 10 4 10 5

10 11 12 13 14 15

FIG. 9. Average of ∆min for different Nf as a function of the filtered FMs d. The scaling of the gap follows ∆min ∝ d−1 which is estimated by a linear regression. The error bars correspond to the first and third percentile of the entire data values on the specific dimension d. The distinct crosses represent the smallest gap ∆∗ , i.e. the hardest instances to simulate.

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E. C. acknowledges funding from the Spanish Ministry for Digital Transformation and the Civil Service of the Spanish Government through the QUANTUM ENIA project call - Quantum Spain, EU, through the Recovery, Transformation and Resilience Plan – NextGenerationEU, within the framework of Digital Spain 2026. A.C.-L. acknowledges funding by the European Union, supported by the EuroHPC Joint Undertaking and its members under the Grant Agreement Nº 101159808, including top-up funding by Ministry for Digital Transformation and the Civil Service of the Spanish Government, and from the grant RYC2022-037769-I funded by MICIU/ AEI/ 10.13039/ 501100011033 and by “ESF+”. F.A.V. and M.A.G.B. acknowledge funding from the Maria de Maeztu Units of Excellence Programme CEX2021-001195-M, funded by MICIU/AEI/10.13039/501100011033. B. J-D acknolwedges funding from PID2023-147475NB-I00 funded by MICIU/AEI/10.13039/501100011033 and FEDER, UE, by grants 2021SGR01095 from Generalitat de Catalunya, and by Project CEX2024-001451-M of ICCUB (Unidad de Excelencia Marı́a de Maeztu).

Nf = 16 Nf = 24 Nf = 32

* Nf = 16 * Nf = 24 * Nf = 32

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FIG. 10. (Left) Plot of ∆min as a function of tmin /τ . For each problem instance, we run the annealing simulation and we identify the fraction of time where the minimum gap ∆min occurs. (Right) Distribution of tmin /τ for all the problem instances. From these results, it can be concluded that the minimum gap is found at the end of the annealing process.

Analysis of overlapping feature maps

In this final section, we present the analysis of the overlap between same subsets of FMs across distinct classes for few data samples. Specifically, we show an example of some of the explanation maps obtained by selecting specific FMs that are shared between distinct classes. Considering the results obtained from the overlap matrix 2, we select the classes airplane, ship and truck, since we noticed a non-zero overlap between them. We extract the selected FMs from the QA for the class airplane for τ = 10, to verify how much variability in selecting different representations the model leads to. In particular, by repeating the FM selection procedure for the same sample, we obtain a global bit string solution containing FMs [0, 1, 15] that is selected most of the time, with variable additional maps that we do not consider here. Thus, we repeat the same procedure for the remaining classes, and we compare the explanation maps with the same subsets of FMs selected.

Class: Airplane

f = [0, 1, 15]

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f = [0, 1, 15]

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FIG. 11. Overlap between subsets of single FMs sampled by the QA for airplane, ship and truck classes. The algorithm is repeated for the same image sample with τ = 10 to verify how much variability in sampling the same bit string the model leads to. From left to right the selected FMs are [0, 1, 15] for the airplane class that we consider as pivots.

The first figure in Fig.11 corresponds to the explanation map containing the global solution sampled by the QA most of the time. From left to right we display the explanation maps corresponding to the single FMs within the selected subset [0], [1], [15]. As we move from left to right, we can clearly see that the model activates FMs that correspond to specific details of the images that is fed with. In particular, by inspecting the airplane explanation maps with the ones created for the ship, we may notice that such representations correspond to parts that are in common between the two classes, for instance the global shape of the airplane we see is similar to the corresponding ship.

Class: Bird

Class: Cat

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QA-filtered GradCAM

GradCAM

GradCAM++

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FIG. 12. Comparison between state-of-the-art methods and the FS-based on QA for some samples of each image class.

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