Decrypting chaotic visual ciphers via quasi quantum neural networks (Q²NNs).

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Learn more: PMC Disclaimer | PMC Copyright Notice Sci Rep . 2026 Mar 11;16:9937. doi: 10.1038/s41598-026-41513-3 Search in PMC Search in PubMed View in NLM Catalog Add to search Decrypting chaotic visual ciphers via quasi quantum neural networks (Q²NNs) Gokul Manavalan Gokul Manavalan 1 Electrical and Computer Engineering Department, Ben-Gurion University of the Negev, Be’er Sheva, 8441405 Israel Find articles by Gokul Manavalan 1, ✉ , Shlomi Arnon Shlomi Arnon 1 Electrical and Computer Engineering Department, Ben-Gurion University of the Negev, Be’er Sheva, 8441405 Israel Find articles by Shlomi Arnon 1 Author information Article notes Copyright and License information 1 Electrical and Computer Engineering Department, Ben-Gurion University of the Negev, Be’er Sheva, 8441405 Israel ✉ Corresponding author. Received 2025 Sep 10; Accepted 2026 Feb 20; Collection date 2026. © The Author(s) 2026 Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ . PMC Copyright notice PMCID: PMC13022186  PMID: 41813725 Abstract We propose a novel Quasi Quantum Neural Network (Q²NN) architecture that integrates classical convolutional networks with variational quantum circuits to decrypt grayscale images encrypted via a multilayer chaotic cryptosystem. This hybrid framework addresses the limitations of classical models when applied to highly nonlinear, permutation-diffusion encrypted data. Q²NN adopts a dual-branch encoder-decoder design, comprising a classical convolutional autoencoder and a variational quantum subnetwork, fused via an adaptive, learnable module that unifies classical and quantum representations. For evaluation, we implement a custom encryption pipeline combining Arnold Cat Map-based spatial permutation, Logistic Map-based pixel-level diffusion, and zigzag chaotic transformations. These operations produce ciphertexts that are key-sensitive, structurally obfuscated, and statistically complex, presenting a significant challenge for conventional decryption models. Leveraging supervised training with known ciphertext-plaintext pairs, Q²NN approximates an inverse mapping from the encrypted domain back to the original image space by co-learning in a joint classical–quantum feature space. Experimental validation on the MNIST dataset demonstrates near-perfect decryption fidelity (MSE < 0.004; SSIM > 0.96), outperforming classical and quantum-only baselines. The underlying chaotic encryption scheme further shows strong cryptographic resilience, with Shannon entropy ≈ 7.96, NPCR > 99.5%, UACI ≈ 33.3%, and negligible spatial correlation. These results highlight the potential of Q²NNs for secure and explainable image decryption, and point to promising directions for hybrid intelligent cryptographic systems in the post-quantum era. Supplementary Information The online version contains supplementary material available at 10.1038/s41598-026-41513-3. Keywords: Quasi Quantum Neural Network (Q²NN), Hybrid Quantum-Classical Learning, Chaotic Cryptography, Arnold Cat Map, Logistic Diffusion, Post-Quantum Visual Decryption, Variational Quantum Neural Networks (VQNN), Secure Visual Regression, Hilbert–Euclidean Fusion, Entropy-based Cipher Metrics Subject terms: Mathematics and computing, Physics Introduction The convergence of quantum computing and machine learning has catalyzed the development of a transformative computational paradigm, quantum machine learning (QML), which seeks to harness the mathematical richness of quantum mechanics to augment the learning capacity of classical algorithms 1 , 2 . Within this framework, quantum neural networks (QNNs) have emerged as powerful architectures capable of performing complex inferential tasks by exploiting quantum entanglement, superposition, and the vast expressiveness of high-dimensional Hilbert spaces 1 , 3 , 4 . These intrinsic quantum properties render QNNs particularly promising for domains requiring high-dimensional data representations, cryptographic inversion, and combinatorial generalization, areas where classical deep learning often encounters structural and representational bottlenecks 5 – 8 . While classical architectures such as convolutional neural networks (CNNs) have demonstrated efficacy in visual pattern recognition and feature abstraction 9 , 10 , their reliance on deterministic, layer-wise operations fundamentally constrains their capacity to model encrypted or non-deterministically transformed data 5 , 6 , 11 . This limitation is particularly acute in vision-based cryptography, where chaotic permutations and nonlinear pixel-level diffusion introduce complexities that overwhelm standard learning mechanisms. As such, recent advances in quantum-classical hybrid architectures offer a compelling route forward, enabling end-to-end trainable systems that capitalize on quantum-enhanced representations while preserving classical scalability 5 , 6 , 11 . In recent years, hybrid architectures combining classical and quantum elements have emerged to address these limitations. These hybrid quantum-classical neural networks integrate quantum circuits into classical deep learning pipelines 12 , 13 , allowing for end-to-end training while preserving computational feasibility. This has enabled proof-of-concept demonstrations in diverse domains including healthcare, finance, and pattern recognition. Several recent works support this growing trend. Min-Gang Zhou et al. 14 , proposed a quantum neural computing model using single-qubit operations and classically-controlled measurements, demonstrating strong classification capabilities with reduced memory and implementation complexity 14 . Similarly, Asel Sagingalieva et al. 5 , employed a deep quantum learning framework to improve drug response prediction in cancer treatment, reporting a 15% performance boost over classical models 5 . Eric Paquet et al. introduced QuantumLeap, a hybrid quantum architecture for time-series forecasting in finance 11 , while Haider T.H. Salim ALRikabi demonstrated QNN-driven face recognition with PCA-based feature extraction using MATLAB-based quantum systems 6 . Recent work has demonstrated the practical potential of hybrid quantum convolutional architectures in medical imaging. Pandey et al. 15 proposed a hybrid quantum–classical CNN with a quantum attention echanism, achieving state-of-the-art skin cancer classification across multiple benchmarks. Similarly, Alqassab et al. 16 introduced a hybrid quantum CNN for multi-class ocular disease diagnosis from fundus images, outperforming established deep learning models. Together, these studies underscore the increasing maturity of hybrid quantum–classical learning and motivate their application to complex inverse and decryption problems involving highly nonlinear transformations. Despite notable progress in quantum machine learning (QML), most hybrid quantum–classical architectures remain focused on classification or narrow-domain regression tasks, offering limited applicability to ill-posed inverse problems such as cryptographic image decryption. These scenarios require models capable of reconstructing semantically meaningful content from highly nonlinear and structurally obfuscated ciphertexts. To address this, we propose a novel hybrid regression framework, the Quasi Quantum Neural Network (Q²NN or QqNN), which integrates classical convolutional neural networks (CNNs) with variational quantum circuits (VQCs) to enable end-to-end decryption of chaotic image encodings. In our setup, encrypted inputs are synthesized via a custom multi-stage chaotic cryptosystem, combining Arnold Cat Map–based permutation 17 , 18 , Logistic Map–driven pixel diffusion 19 , 20 , and zigzag reordering 21 , 22 , resulting in ciphertexts with minimal spatial coherence and near-random statistical structure. The Q²NN is trained under a supervised setting with known ciphertext–plaintext pairs, learning a continuous inverse mapping from the encrypted domain to the original image manifold using a joint classical–quantum representation. Unlike sequential hybrid models, Q²NN adopts a parallel fusion topology, wherein quantum circuits capture high-dimensional abstract correlations, while CNNs extract spatial priors, with their outputs dynamically fused via a learnable entropy-aware module. Evaluated on the MNIST dataset, Q²NN achieves high decryption fidelity (MSE and SSIM were as low as 0.003 and > 0.96, respectively), reducing reconstruction error by up to 89.2% and improving perceptual similarity by 20.6% relative to standalone CNNs. Simultaneously, the chaotic encryption pipeline demonstrates strong cryptographic robustness, with entropy values nearing 8, inter-pixel correlations near zero, and NPCR/UACI metrics close to theoretical optima. These results position Q²NN as a promising prototype for post-quantum cryptanalysis and secure visual systems in adversarial and information-obscured environments. Materials and methods This section delineates the architectural formulation and algorithmic components underpinning the proposed Q²NN, a hybrid variational model for decrypting chaotically obfuscated grayscale image data. The framework is designed to address the limitations of purely classical architectures when confronted with permutation-diffusion encrypted visual inputs. The overall structure integrates a dual-branch encoder-decoder architecture: a classical convolutional autoencoder and a quantum variational subnetwork, both of which operate in parallel and are unified via an adaptive learnable fusion module. A schematic overview is depicted in Figs. 1 and 2 . While classical CNNs provide robust local feature extraction via translational weight sharing 9 , 10 , they struggle with learning inverse mappings over highly nonlinear, globally permuted data. In contrast, VQCs offer superior representational flexibility through non-Euclidean embeddings, entanglement-based correlation modeling, and the ability to navigate exponentially large Hilbert spaces 6 . The Q²NN framework synthesizes these paradigms into a cohesive end-to-end differentiable architecture, trained under shared supervision to recover high-fidelity image representations from structurally disordered encrypted domains. All simulations and training were conducted on Google Colab using PyTorch for the classical components and PennyLane for quantum circuit simulation 23 , 24 . Algorithm 1 outlines the complete encryption-decryption pipeline using the Q²NN. Pseudocode for the Q²NN is provided in Supplementary Section S1, followed by an explanation of its significance in Section S2 of the same document. The quantum backend used was default.qubit 25 , which supports full-state vector simulation, enabling operations such as angle embedding, entanglement via RX/RY gates, and Pauli-Z measurements 26 . This setup allows us to emulate key quantum behaviors, such as superposition and entanglement, within a classical computing environment, forming the foundation for the proposed quasi-quantum learning pipeline. The more detailed and descriptive implementation section is explained in the supplementary section S3 and S4. Fig. 1. Open in a new tab Structured flowchart of the proposed chaotic encryption and hybrid Q²NN decryption framework. The encryption stage (left) applies Arnold Cat Map permutation, logistic-map-based diffusion with feedback, and zigzag scrambling to generate the ciphertext. The decryption stage (right) processes the encrypted image through parallel classical (CNN autoencoder) and variational quantum branches, whose outputs are adaptively fused and optimized to reconstruct the original image. Fig. 2. Open in a new tab Schematic overview of the proposed quasi-quantum neural network (Q²NN) architecture: The encrypted grayscale input image is simultaneously processed along two independent pathways: a classical CNN branch and a QNN branch. Each branch reconstructs a decoded version of the input via separate decoder modules. The outputs of both branches are fused using a learnable fusion block, yielding the final reconstructed image. A loss block computes the combined error against the ground truth, backpropagating gradients jointly through both branches for end-to-end training. Algorithm 1. Open in a new tab Chaotic Encryption and Reconstruction Using Q²NN. Chaotic encryption protocol To emulate a cryptographic decryption scenario, we employ a compound chaotic encryption pipeline designed to obscure spatial, statistical, and perceptual patterns in grayscale images. The encryption process incorporates deterministic permutation (Arnold Cat Map) 17 , 18 , chaotic diffusion (Logistic Map) 19 , 20 , and Zigzag scanning 21 , 22 , thereby ensuring that both pixel-wise and structural statistics of the original image are obfuscated. Let denote the normalized grayscale image for sample i . The encryption proceeds as follows: Arnold Cat Map-Based Permutation . The Arnold Cat Map (ACM) is utilized to permute the pixel positions of the input grayscale image. Given an image of size , the pixel at location is remapped to a new location using 17 , 18 : 1 Where are positive integers controlling the mixing dynamics and are the spatial coordinates. ensures periodic boundary conditions on the pixel domain with N representing image width ( N = 28). Multiple permutation iterations can be applied to enhance the spatial entropy. The ACM exhibits periodic yet pseudo-random behavior with high sensitivity to initial parameters, making it ideal for pixel-level scrambling 17 , 18 . The permuted image is subsequently flattened into a 1D sequence for diffusion. b) Pixel-Value Diffusion via Logistic Map and zigzag scan reordering . After spatial permutation with the Arnold Cat Map, the pixel intensities are diffused using a chaotic feedback-based XOR mechanism driven by the Logistic Map 19 , 20 . The one-dimensional Logistic Map sequence is defined as: 2 Where is the n th element of the chaotic sequence, and ensures chaotic behavior 19 , 20 . For each permuted pixel , the diffusion is computed iteratively: 3 Where is the encrypted pixel value at position i , denotes the bitwise XOR operation, and is the scaled Logistic Map sequence for diffusion. The previously encrypted pixel is introduces inter-pixel dependency (feedback), while the modulo operation ensures the output remains an 8-bit integer. This iterative feedback ensures that small changes in the input image or key propagate throughout the ciphertext, increasing diffusion strength and enhancing resistance to differential attacks. After the iterative XOR diffusion, the encrypted pixels are reordered using a zigzag traversal to disrupt spatial adjacency and enhance keyspace entropy 21 , 22 . Let the zigzag operator be denoted as ; the final encrypted sequence is then: . This operation ensures that neighboring pixels in the permuted and diffused matrix are not sequentially aligned in the ciphertext, strengthening both statistical and structural security of the encrypted image. c) Key Generation Specification . For each encryption instance, the secret key comprises the parameters controlling both the permutation and diffusion stages. Arnold Cat Map (ACM) permutation: p , q ∈ {1,2,3,4,5} are drawn uniformly at random, and the number of permutation iterations is drawn uniformly from 5 to 20. These parameter ranges follow established practices to maintain bijective pixel shuffling and strong spatial mixing 17 , 18 . Logistic Map diffusion: The bifurcation parameter µ ∈ [3.9,4.0] and initial condition z 0 ∈ [0,1] are drawn uniformly at random. The sequence generated from the Logistic Map is used to diffuse the permuted pixels, with the sequence length equal to the number of pixels. No additional iteration parameter is randomly sampled for the Logistic Map. This choice ensures chaotic dynamics required for strong diffusion 27 – 29 . Each key tuple is logged for reproducibility. By explicitly specifying the ranges and distributions, the encryption procedure becomes fully reproducible while preserving the unpredictability and high entropy necessary for secure chaotic encryption. Dataset and preprocessing We utilize the standard MNIST dataset 30 , where each input image represents a grayscale digit with corresponding label . To formulate a class-conditioned encrypted reconstructed task, we restrict training to a single class . The remaining classes follow the same procedure to perform the proposed methodology, enabling deterministic encryption-decryption mapping. Each image is normalized: 4 The encrypted image is generated using the procedure in Sect. 2.1. The model learns a mapping from to , optimized for reconstruction fidelity. Decryption framework The chaotic encryption process can be formally described as a nonlinear bijective operator: parameterized by a secret key set , where, define Arnold Cat Map permutation, define the logistic diffusion generator. Given a plaintext image , the ciphertext is: . The decryption task is formulated as learning a parametric inverse mapping: , Where denotes trainable parameters of the proposed hybrid network. Unlike analytical inversion, our approach learns a data-driven approximation of the inverse chaotic operator. Classical decryption branch (CNN Autoencoder) The classical branch implements a nonlinear encoder–decoder mapping: : to , : to . The latent representation is 31 and reconstruction: 5 Here, d = 128 is the latent dimension, chosen based on empirical convergence behavior, as shown in Sect. 3.3.b. Interpretation as Learned Inverse Diffusion : Chaotic encryption introduces spatial permutation via the Arnold Cat map (ACM) and sequential diffusion as mentioned in Sect. 2.1. Then the CNN learns to recover the spatial coherence disrupted by permutation, approximate the recursive inverse of the diffusion process, and restore local intensity dependencies. However, convolutional filters primarily capture local receptive fields, which limits their ability to model long-range chaotic correlations. Variational Quantum Branch (QNN) The quantum branch processes a compressed feature vector: , n = 16. The simulated quantum system evolves in 32 . The embedding of data into quantum space proceeds as follows: State Initialization: 6 This represents the quantum system initialized in the ground state, with all qubits set to zero. 2. Angle Embedding 33 : 7 Each component of the pre-encoded classical vector is used to rotate a qubit around the Y-axis via the gate. This process places each qubit into a superposition, allowing the quantum state to represent multiple possible values simultaneously 25 , 26 . 3. Trainable entangling evolution. 8 A trainable unitary operation is applied to introduce entanglement—a uniquely quantum phenomenon in which the state of one qubit becomes intrinsically correlated with others. In our setup, this is implemented using BasicEntanglerLayers 34 , with parameterized and gates acting across qubit pairs. Entanglement enables the circuit to model complex, non-local correlations that would be difficult or inefficient to capture in purely classical networks 25 , 26 . 4. Measurement projection. 9 After entanglement, each qubit is measured in the Pauli-Z basis, yielding an expectation value that reflects the probability-weighted outcome of observing the qubit in the ∣0⟩ or ∣1⟩ state 26 . These expectation values form a feature vector , which retains information about the quantum-evolved structure of the encrypted data. Finally, the quantum-encoded vector is passed through a classical fully connected decoder (Fig. 2 ) network to generate a reconstructed image from the QNN path: 10 Functional Interpretation : The quantum circuit performs a parameterized nonlinear unitary transformation: , which induces a non-Euclidean embedding of encrypted data before classical decoding. This does not assume quantum advantage, but rather introduces an alternative nonlinear transformation prior. Adaptive fusion (Q²NN or QqNN) The final reconstruction combines the CNN output ( ) and the QNN output ( ) through a learnable fusion weighting mechanism: 11 Here, is a trainable scalar optimized during backpropagation. This approach captures complementary features from both quantum and classical embeddings, improving reconstruction robustness. The architecture is trained end-to-end, with shared supervision. The final decryption output is obtained by adaptively fusing the CNN-based and QNN-based reconstructions. The learnable fusion parameter α balances spatially local reconstruction from the classical branch with globally correlated features extracted by the quantum branch. This adaptive fusion allows the model to dynamically prioritize the most informative representation during decryption, improving robustness across different encryption keys and chaotic configurations. To provide a consolidated overview of the complete cryptographic learning pipeline, Fig. 1 presents a structured flowchart integrating both encryption and decryption processes. The left portion of the diagram illustrates the chaotic encryption stages, including Arnold Cat Map permutation, logistic-map-based diffusion with feedback, and zigzag reordering. The right portion depicts the hybrid decryption framework, where the encrypted image is processed through parallel CNN and quantum branches, followed by adaptive fusion and optimization to reconstruct the original image. This visual representation complements the formal mathematical formulation presented in Sect. 2.1–2.3 and enhances procedural clarity. Theoretical basis and representational advantage of Q²NN The quasi-quantum neural network (Q²NN) extends classical feature learning by embedding encrypted data into a complex Hilbert space , where n denotes the number of qubits. Each encrypted vector is encoded by a parameterized unitary operator , defining a nonlinear feature map: 12 The expectation values measured on this quantum state, 13 represent a nonlinear transformation of the input within an exponentially large complex space. The inner product between two quantum states, 14 defines a quantum kernel whose expressive capacity can surpass that of polynomial-depth classical models 35 – 37 . For decryption of chaotic visual data, where permutation–diffusion processes destroy spatial locality, this mapping allows the model to capture global statistical correlations between scrambled pixels. The entanglement operations within effectively impose a non-local coupling structure analogous to long-range receptive fields, enabling the network to learn inverse mappings of complex chaotic transformations. Consequently, the Q²NN combines the local texture extraction ability of CNNs with the global correlation modeling of quantum embeddings, offering a representational synergy unattainable by purely classical architectures 35 – 37 . Furthermore, although scalable quantum computers are not yet widely accessible, the proposed framework is designed to be hardware-agnostic and forward-compatible with emerging quantum accelerators, positioning Q²NN as a practical bridge toward future high-speed quantum-assisted decryption systems. From a cryptographic perspective, the decryption task corresponds to learning an inverse mapping of a nonlinear chaotic operator composed of permutation and diffusion stages. The quantum embedding defines a nonlinear feature map whose induced kernel operates in an exponentially large space, enabling efficient approximation of inverse mappings that are difficult to represent using polynomial-depth classical networks. This theoretical perspective clarifies why the proposed Q²NN is particularly effective for chaotic image decryption. Loss function and optimization strategy To optimize both pixel fidelity and perceptual quality, we use a composite loss function combining MSE and SSIM metrics: 15 Where and SSIM is the structural similarity index. balances MSE and perceptual quality. The optimization process employs the Adam optimizer, set with a learning rate of 3 × 10 − 3 and a batch size of 10. Training proceeds for a maximum of 40 epochs, with early stopping based on validation loss and a patience of 10 epochs. The model was implemented in Python using PyTorch and PennyLane 23 . Training and evaluation were performed on Google Colab using GPU acceleration. The quantum simulations used the default.qubit backend for emulation of quantum circuits. All images were visualized using Matplotlib, and metrics were computed using Scikit-learn and pytorch-msssim 38 , 39 . Code availability All methodological code, dataset preprocessing, and encryption pipelines used in this study are publicly available in the Zenodo repository (DOI: 10.5281/zenodo.18681790). The repository is organized into five modular Jupyter notebooks corresponding to the experimental workflow: environment setup, dataset encryption, Q²NN hybrid model implementation, training strategy, and evaluation. The notebooks include all scripts required to reproduce the figures, tables, and results reported in the manuscript. Execution was performed on Google Colab with fixed random seeds and deterministic parameter initialization where feasible. Minor numerical deviations may occur due to stochastic initialization in quantum layers and simulator variability, but overall trends and comparative performance of the QNN and hybrid models are fully reproducible. Results This section presents a rigorous evaluation of the proposed Q²NN in comparison to classical convolutional neural networks CNNs and standalone QNNs. All models are assessed on their capacity to decrypt images encrypted through a multi-layer chaotic pipeline. The evaluation employs a suite of perceptual, statistical, and cryptographic metrics to characterize both model performance and cipher resilience. Training dynamics and convergence behavior We analyzed the training dynamics of CNN, QNN, and hybrid Q²NN models across digit classes 1, 5, and 8 from the MNIST dataset (Fig. 3 a–c). Each class corresponds to images of a specific numeral, allowing us to test model robustness across varying digit geometries 24 . The Q²NN was optimized using a composite loss function that combines mean squared error (MSE) with the structural similarity index (SSIM), encouraging both perceptual fidelity and pixel-wise accuracy. The superior convergence behavior of Q²NN relative to the quantum-only branch can be attributed to the stabilizing influence of the classical CNN component. In the hybrid architecture, CNN layers capture robust spatial priors and propagate high-magnitude gradients during backpropagation. These stable gradients are shared with the quantum variational branch through the learnable fusion coefficient α, effectively mitigating the well-known “barren plateau” phenomenon in variational quantum circuits (VQCs), which is characterized by vanishing gradients and unstable optimization 40 . As a result, the QNN component can optimize more effectively, leading to smoother loss trajectories and lower terminal losses across all digit classes. Fig. 3. Open in a new tab Training loss curves for ( a ) Class 1, ( b ) Class 5, and ( c ) Class 8: The plot shows the evolution of the combined loss (MSE + SSIM) across 40 epochs for CNN, QNN, and the fused Q²NN (or QqNN) model. The Q²NN demonstrates a consistently lower loss and more stable convergence compared to the individual models. For instance, in Class 1, the CNN reduced its initial loss from 0.3208 to 0.1388, while the QNN-only branch achieved a final loss of 0.1278, albeit with oscillatory behavior symptomatic of barren plateaus. In contrast, Q²NN initialized at a lower loss (0.1264) and converged steadily to 0.0978, highlighting its enhanced gradient flow and representational richness. Similar trends were observed for Classes 5 and 8, where Q²NN outperformed both baselines with 15–30% lower terminal loss values. These findings align with recent literature demonstrating that hybrid quantum-classical architectures, where classical layers provide stable gradient flows, can significantly alleviate barren plateau effects and improve convergence in VQCs 40 . Overall, the Q²NN architecture effectively integrates the localized convolutional features captured by CNNs with the abstract entangled embeddings produced by the quantum branch. The dynamic, learnable fusion facilitates stable optimization, improves convergence, and enhances reconstruction fidelity even for highly obfuscated ciphertexts. Behavior of the fusion parameter α In Eq. ( 11 ) we introduced the learnable scalar α ∈ [0,1] that adaptively weights the classical and quasi-quantum branch outputs. To assess the relative contributions of each branch we monitored α throughout training on the MNIST reconstruction/decryption experiments. α was initialized at 0.50 and the model was trained for 40 epochs under the same training regime reported in Fig. 3 . The observed trajectory of α was smooth and convergent (Fig. 4 ): epoch 10 → 0.63, epoch 20 → 0.68, epoch 30 → 0.70, epoch 40 (converged) → 0.72. Fig. 4. Open in a new tab Convergence of the learnable fusion parameter α over 40 training epochs. The final learned value (𝛼 ≈ 0.72) indicates that the classical branch accounts for approximately 72% of the fused output, while the quantum branch contributes ~ 28%. On this low-dimensional task (MNIST), the classical CNN therefore dominates, with the quantum embedding acting as a complementary feature extractor. Importantly, the smooth convergence of 𝛼 demonstrates the stability of the hybrid training process. Monitoring 𝛼 thus provides interpretability of branch contributions in our Q²NN architecture. In more complex or higher-dimensional scenarios, we anticipate that 𝛼 might decrease (i.e., shift weighting toward the quantum branch) if the classical embedding alone becomes insufficient—making 𝛼 a useful empirical indicator of modality balance. Similar learnable fusion-weight mechanisms have been reported in multimodal learning literature, including Wang et al. 41 and Md. Rumman Rafi et al. 42 , where variable-weight strategies are used to adaptively allocate representational capacity among modalities. Architectural rationale and ablation summary The architectural parameters of the proposed hybrid quantum–classical network were determined through empirical analysis and supported by current literature. Two key design aspects were evaluated: (i) the number of qubits ( n ) in the variational quantum circuit, and (ii) the latent-space dimensionality ( d ) of the classical encoder. Qubit optimization . We conducted an ablation sweep over n = {4,8,12,16,20} using the MNIST dataset (28 × 28 pixels). Figure 5 presents a four-panel analysis of model performance and computational scaling. Subplots (a) and (b) show that test accuracy and validation loss improve steadily up to n = 16 qubits, achieving peak accuracy and minimum loss. Subplots (c) and (d) demonstrate that runtime and GPU memory usage grow super-linearly with circuit width, with a pronounced increase beyond n = 16. The optimal configuration at n = 16 therefore offers a balanced compromise between predictive power and computational efficiency. These results are consistent with recent findings that the number of qubits required for effective hybrid learning typically lies within the low-teen range under current NISQ hardware limitations 43 . Fig. 5. Open in a new tab Ablation Study of Quantum Circuit Width (Qubit Count): This four-panel analysis examines the impact of increasing the number of qubits ( n ) on model performance and computational cost. Subplots ( a ) and ( b ) track performance metrics, while subplots ( c ) and ( d ) measure resource consumption. (a) Test Accuracy (%) shows improvements up to n = 16 qubits, beyond which gains plateau. (b) Validation Loss (a.u.) exhibits a corresponding minimum near n = 16. (c) Time per Epoch (s) and (d) GPU Memory Used (MB) both demonstrate non-linear increases in computational overhead with circuit scaling. b) Latent-space dimension . The latent dimension d of the classical encoder was tuned among {64,128,256}. As shown in Fig. 6 , Empirically, d = 128 provided stable convergence behavior, avoided over-compression, and maintained manageable model complexity. Expanding to d = 256 increased computational cost without significant improvement in accuracy. This observation agrees with the conclusions of Ma et al. (2024), who demonstrated that compact latent embeddings in hybrid quantum autoencoders enhance stability and facilitate more efficient quantum encoding 44 , 45 . Fig. 6. Open in a new tab Ablation Study of Classical Encoder Latent-Space Dimension: This four-panel analysis examines the effect of latent-space dimension ( d ) on hybrid model performance and computational efficiency. ( a ) Test Accuracy (%) increases from d = 64 to d = 128, then saturates at d = 256, indicating diminishing returns. ( b ) Validation Loss (a.u.) shows a similar trend, reaching its minimum at d = 128. ( c ) Time per Epoch (s) (plotted on a logarithmic scale) and (d) GPU Memory Usage (MB) both rise substantially with larger latent spaces, reflecting the added cost of higher-dimensional encoders. These results support the choice of d = 128 as an optimal trade-off between model performance and resource efficiency. Quantitative performance evaluation across models Table 1 summarizes the quantitative decryption performance of the proposed Q²NN framework in comparison with classical CNN and standalone QNN baselines across three representative MNIST digit classes (1, 5, and 8). To ensure a fair and comprehensive evaluation, we report multiple complementary metrics widely used in image reconstruction and inverse imaging literature, including Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Peak Signal-to-Noise Ratio (PSNR), and Structural Similarity Index (SSIM) (e.g., based on established evaluation practices in medical image synthesis and inverse problems) 46 – 48 . While MSE and RMSE quantify pixel-wise reconstruction fidelity, MAE provides a robust measurement of absolute pixel intensity deviations. PSNR evaluates the relative signal quality and noise suppression on a logarithmic decibel scale. SSIM captures perceptual and structural similarity by comparing luminance, contrast, and structural features between reconstructed and reference images. Table 1. Quantitative performance comparison of CNN, QNN, and Q²NN across target classes on encrypted image reconstruction (Mean ± Std). Metric Model Class 1 Class 5 Class 8 MSE Q²NN 0.0037 ± 0.0006 0.0052 ± 0.0007 0.0026 ± 0.0008 CNN 0.0280 ± 0.0006 0.0209 ± 0.0009 0.0251 ± 0.0005 QNN 0.0130 ± 0.0010 0.0244 ± 0.0009 0.0163 ± 0.0007 RMSE Q²NN 0.0567 ± 0.0010 0.0695 ± 0.0006 0.0594 ± 0.0007 CNN 0.1691 ± 0.0007 0.1453 ± 0.0007 0.1596 ± 0.0015 QNN 0.1128 ± 0.0009 0.1588 ± 0.0008 0.1286 ± 0.0006 MAE Q²NN 0.0433 ± 0.0006 0.0538 ± 0.0006 0.0454 ± 0.0004 CNN 0.1360 ± 0.0007 0.1161 ± 0.0014 0.1272 ± 0.0008 QNN 0.0899 ± 0.0011 0.1270 ± 0.0011 0.1024 ± 0.0008 PSNR (dB) Q²NN 24.8888 ± 0.0103 23.1907 ± 0.0094 24.5581 ± 0.0090 CNN 15.4193 ± 0.0039 16.7822 ± 0.0108 15.9522 ± 0.0092 QNN 18.9355 ± 0.0091 15.9877 ± 0.0053 17.8295 ± 0.0107 SSIM Q²NN 0.9639 ± 0.0014 0.9406 ± 0.0017 0.9576 ± 0.0014 CNN 0.8264 ± 0.0012 0.7596 ± 0.0018 0.8141 ± 0.0032 QNN 0.8866 ± 0.0010 0.7089 ± 0.0011 0.8537 ± 0.0031 Open in a new tab Across all classes, the Q²NN model consistently achieves the lowest reconstruction errors. In terms of RMSE, Q²NN obtains values in the range of 0.055–0.069 and MAE values below 0.055 across all classes, indicating highly accurate pixel-wise recovery. CNN and QNN models exhibit substantially higher RMSE and MAE, reflecting residual blurring and localized discrepancies in their reconstructions. From a signal fidelity perspective, Q²NN attains PSNR values between 23.19 dB and 25.09 dB, significantly outperforming both CNNs and QNNs. Similarly, the perceptual quality measured by SSIM remains consistently high for Q²NN (SSIM > 0.94), whereas CNN and QNN baselines show lower structural similarity. To ensure robustness and reproducibility, results for each model are averaged over multiple independent training runs with different random initializations. The Q²NN framework exhibits lower variance across runs compared with the QNN baseline, indicating improved optimization stability and reduced sensitivity to initialization. It is important to note that these results are obtained on the MNIST dataset, which consists of low-resolution (28 × 28) grayscale images. Within this constrained setting, the achieved error levels, PSNR, and SSIM values approach practical upper bounds for encrypted image reconstruction. For higher-resolution and information-rich images such as natural or medical imagery, we anticipate that the relative advantages of the proposed Q²NN framework will become even more pronounced, as complex feature correlations across larger latent spaces can be more effectively captured. Cryptographic robustness of chaotic encryption To validate the strength of the encryption layer, we evaluated statistical measures commonly used in cryptographic security analysis. Table 2 summarizes Shannon entropy 49 , directional correlation coefficients (horizontal, vertical, diagonal) 49 , 50 , NPCR (Number of Pixels Change Rate) 51 , 52 , and UACI (Unified Average Changing Intensity) for encrypted images across the three digit classes 51 , 52 . The encryption protocol, composed of Arnold Cat Map–based spatial permutation, Logistic Map–based diffusion, and chaotic zigzag ordering—yielded ciphertexts with near-ideal properties. Table 2. Statistical metrics for analyzing the robustness of the encryption performance. Metric Target class 1 Target class 5 Target class 8 İnce et al 53 . Kumar et al 54 . Ge et al 55 . Ideal Value Shannon Entropy 7.9521 7.9634 7.9587 7.99 7.99 7.99 8 Correlation (H) 0.0058 0.0021 −0.0039 - - - 0 Correlation (V) 0.0084 −0.0065 0.0051 - - - 0 Correlation (D) 0.0063 −0.0019 0.0072 - - - 0 NPCR (%) 99.5213 99.5187 99.5099 99.6122 99.5875 99.6094 99.6094 UACI (%) 33.2845 33.2109 33.2987 33.4690 33.4121 33.4635 33.4635 Open in a new tab Shannon entropy values approach the theoretical maximum of 8 for all target classes, while horizontal, vertical, and diagonal correlation coefficients remain near zero, confirming effective decorrelation of neighboring pixels. NPCR and UACI metrics further verify strong diffusion and high sensitivity to pixel-level perturbations, demonstrating robustness against differential attacks. Unlike many hybrid quantum–classical learning studies, this work explicitly integrates cryptographic benchmarking, making it one of the first to assess both decryption performance and encryption resilience within a unified framework. Figure 7 depicts the corresponding histograms: the original images exhibit non-uniform distributions with distinct peaks, whereas the encrypted images display nearly uniform distributions across the 0–255 intensity range—a visual confirmation of high entropy and decorrelation. Fig. 7. Open in a new tab Histograms demonstrating the cryptographic effectiveness of the chaotic encryption scheme on grayscale images. ( a ) An original (unencrypted) digit image; ( b ) The histogram of the original image, showing a non-uniform pixel distribution; ( c ) The corresponding encrypted image; ( d ) The histogram of the encrypted image, displaying a uniform, flat distribution. To contextualize the difficulty of the decryption task, Table 2 presents the cryptographic performance of the proposed Arnold–Logistic–Zigzag encryption scheme alongside contemporary chaotic encryption methods. Our method achieves comparable or superior NPCR, UACI, and entropy values, validating its strength as a benchmark cipher and underscoring the necessity of the proposed Q²NN for accurate recovery of heavily obfuscated ciphertexts. Minor variations relative to reported benchmarks arise from dataset differences and the limited dimensionality of MNIST (28 × 28 grayscale), which modestly affect entropy and diffusion statistics without altering the overall cryptographic strength of the scheme. Visual reconstruction quality and perceptual fidelity To complement the quantitative results, we present visual comparisons of decrypted outputs across all three models (Fig. 8 ). CNN-based reconstructions retained general shape but failed to recover fine-grained edge details, while QNN outputs captured abstract structure yet lacked pixel-level precision. Q²NN-generated reconstructions closely matched the original images in both global morphology and local texture, preserving digit boundaries and internal symmetry with high fidelity. The dual-branch architecture enabled Q²NN to fuse localized convolutional features with globally entangled quantum embeddings. This fusion, combined with SSIM-driven perceptual optimization, facilitated photorealistic decryption, even from heavily obfuscated ciphertexts. The qualitative clarity observed in Q²NN outputs further corroborates the numerical improvements in MSE and SSIM. Fig. 8. Open in a new tab Qualitative decryption results per class: Columns ( a ) Ground truth digit image (unencrypted); ( b ) Encrypted input image using a multilayer chaotic cryptosystem; ( c ) Reconstructed image using CNN; ( d ) Reconstructed image using QNN; ( e ) Final output from Q²NN fusion model. The fusion output shows visually superior decryption quality with better edge definition and structural fidelity. Discussion The Q²NN framework represents a significant advancement in quantum-assisted image decryption by treating decryption as a supervised inverse modeling problem, rather than a deterministic cipher reversal. By integrating classical convolutional networks with VQCs, Q²NN bridges the gap between spatial structure capture and global nonlocal pattern recognition in high-entropy, non-invertible chaotic encryption schemes 1 , 3 , 5 . Through a parallel dual-branch design and an entropy-aware fusion mechanism, it achieves robust decryption of structurally obfuscated images, with performance exceeding classical and quantum-only baselines (MSE ≈ 0.0031; SSIM > 0.96). A comparative overview of recent quantum and chaos-based image encryption schemes is presented in Table 3 , which highlights Q²NN’s novel contribution, specifically, its trainable decryption model that generalizes across encrypted image distributions. Table 3. Comparative analysis of recent quantum and chaotic image encryption methods against the proposed Q²NN framework. Study Encryption Method Decryption Approach Quantum Involvement Main Advantage Key Limitation Wang et al. (2022) 56 Time-delay; complex chaotic network Classical deterministic None Large key space, good statistical defense No learning-based decoding Liu et al. (2024) 57 GQIR; 4D chaos; Arnold Classical inverse mapping Used for encryption only High entropy, low correlation No inference model; resource-heavy Shi et al. (2020) 58 CV-QNN-based cipher Quantum CV neural net Quantum for all phases Full quantum cryptosystem Limited to simple data, hardware limits Hao et al. (2023) 59 Quantum walk; XOR; NEQR Deterministic decryption Full quantum Adaptive quantum ops, novel PRNG No trainable recovery mechanism Wu et al. (2025) 60 QLSTM-enhanced chaotic encryption Sync-based decryption Quantum-enhanced chaos Strong attack resistance, sync control Requires sync; lacks generalization Shafique et al. (2024) 61 QKD; hyperchaos; DCT/DWT Classical; quantum keys Quantum key exchange only Strong multi-layer security, IoT focus Encryption-focused; complex pipeline Ours (Q²NN) Arnold; Logistic; Zigzag chaotic layers Trainable hybrid quantum-classical decoder VQCs in training and inference Learns inverse map of non-invertible ciphers; NISQ-compliant Grayscale only; fusion module lacks interpretability Open in a new tab Recent work has largely focused on designing increasingly complex encryption pipelines, leveraging chaotic maps, multi-stable networks, or quantum-inspired entropy sources 56 – 61 . Wang et al. (2022) 56 , proposed a time-delay chaotic encryption scheme with a large key space and good statistical robustness, but relied entirely on classical synchronization and lacked any data-driven decryption component. In contrast, Q²NN learns to invert encrypted images through supervised learning, which avoids dependence on synchronized chaotic systems or deterministic key matching. Liu et al. (2024) 57 , employed a 4D chaotic map with generalized quantum image representations (GQIR), achieving high entropy and decorrelation. However, their quantum involvement was limited to the encryption process, and decryption was deterministic and invertibility-dependent. Q²NN, by contrast, utilizes quantum circuits during training and inference to learn flexible inverse mappings that can handle nonlinear and non-invertible encryption scenarios. In the quantum learning domain, Shi et al. (2020) 58 , presented a continuous-variable quantum neural network (CV-QNN) encryption scheme for low-dimensional data. Though their work represents a foundational step in quantum cryptography, it is not scalable to structured image data and incompatible with current NISQ hardware due to the use of continuous variables. Q²NN sidesteps this by employing shallow, discrete qubit-based VQCs, enabling realistic deployment in near-term quantum systems. Hao et al. (2023) 59 , leveraged quantum walks and quantum XOR operations for encryption but relied on deterministic decryption and assumed ideal quantum operations. Q²NN addresses this limitation by learning the decryption function through supervised training using plaintext-ciphertext pairs. While key knowledge is essential during training to generate encrypted data, Q²NN does not require explicit key input or handcrafted cipher inversion rules at inference. This distinction allows generalization across encrypted inputs without knowing the underlying cipher structure. Wu et al. (2025) 60 , proposed a QLSTM-based chaotic system with strong synchronization-based decryption and high robustness. However, such schemes rely on sender-receiver synchrony, limiting flexibility in dynamic or adversarial networks. Q²NN learns a generalizable decryption function and is inherently asynchronous, requiring only encrypted inputs at inference. Shafique et al. (2024) 61 , addressed security in medical IoT by combining QKD, DCT/DWT, and chaotic scrambling. Although highly secure, their approach is heavily encryption-focused and lacks a trainable decoder that can operate under structural uncertainty. Q²NN directly tackles this challenge by framing chaotic decryption as a supervised learning task. It is important to emphasize that the chaotic encryption pipeline employed in this work is not proposed as a novel cryptographic cipher. Rather, it serves as a high-entropy benchmark designed to evaluate the ability of the proposed Q²NN framework to learn inverse mappings under severe structural obfuscation. The primary contribution lies in the decryption paradigm—formulating chaotic decryption as a supervised inverse learning problem—rather than in the encryption construction itself. Additional implementation details and potential application scenarios are provided in the Supplementary Material (Section S5). Recent advances have further demonstrated the promise of quantum neural networks in optical wireless communication, particularly for compensating distortions in orbital angular momentum (OAM) beams propagating through multiphysics channels 62 . These developments motivate the extension of the proposed Q²NN framework toward secure OAM-based optical communication systems, where learning-based decryption and reconstruction under complex channel impairments remain open challenges. Conclusion and future work In this work, we introduced Q²NN, a hybrid quantum–classical framework for supervised learning-based decryption of chaotic images. By integrating CNNs with VQCs and employing an entropy-aware fusion mechanism, Q²NN effectively learns a flexible inverse mapping from highly obfuscated encrypted images to their plaintext counterparts. The proposed framework overcomes key limitations of prior approaches, including reliance on deterministic cipher inversion, synchronization dependency, and explicit key recovery. Quantitative evaluation on the MNIST dataset demonstrates that Q²NN significantly outperforms classical CNN and standalone QNN baselines across multiple complementary metrics. Specifically, Q²NN achieves a MSE of 0.0032 ± 0.0006, a RMSE of 0.0567 ± 0.0010, a MAE of 0.0433 ± 0.0006, a PSNR of 24.89 ± 0.01 dB, and a SSIM of 0.9639 ± 0.0014. These results confirm that Q²NN delivers highly accurate pixel-wise recovery, preserves structural integrity, and maintains robustness across independent training runs. The dual-branch architecture enables the classical CNN to capture local spatial patterns while the quantum branch models global, non-local correlations induced by chaotic encryption, and the entropy-aware fusion balances these complementary contributions, providing enhanced decryption fidelity. From a conceptual perspective, Q²NN frames chaotic decryption as a supervised inverse problem rather than deterministic key reversal. Unlike existing chaos- or quantum-based schemes, our approach generalizes across encrypted distributions without requiring explicit cipher knowledge, synchronized chaotic systems, or idealized quantum operations. Consequently, the proposed framework establishes a scalable, learning-driven paradigm for post-quantum cryptanalysis, inverse imaging, and secure communications applications. Building on these results, future research will focus on extending Q²NN to higher-resolution and more complex data domains. This includes implementing quantum feature compression, dimensionality reduction, and tensor-network–inspired hybrid circuits to enable efficient decryption of large-scale images while controlling qubit requirements and circuit depth. Optimization of variational circuits under noisy intermediate-scale quantum (NISQ) constraints, including pruning, error mitigation, and embedding strategies, will further enhance performance and reliability on near-term hardware 63 , 64 . In parallel, the interpretability of the entropy-aware fusion mechanism will be investigated to understand the respective contributions of classical and quantum branches, which is critical for high-stakes applications such as medical imaging or defense-oriented decryption. Robustness against quantum-enabled adversarial threats, including Grover-assisted search strategies 65 , model inversion attacks, and other emerging quantum adversarial scenarios, will also be explored. Extensions to color, hyperspectral, and multimodal data will broaden applicability to richer information domains, and integration into secure optical and orbital angular momentum (OAM) communication systems will leverage quantum-enhanced reconstruction for error-resilient recovery under complex channel distortions. Overall, Q²NN provides a principled and high-performance framework for learning-based chaotic decryption, offering a path toward robust, explainable, and scalable inverse mapping under high-entropy, non-invertible transformations. As hybrid quantum–classical technologies continue to mature, this framework holds promise for broader deployment in secure communications, inverse imaging, and post-quantum intelligent systems, providing a scientifically rigorous foundation for next-generation learning-driven cryptanalysis and reconstruction tasks. Supplementary Information Below is the link to the electronic supplementary material. Supplementary Material 1 (93.1KB, docx) Acknowledgements This work was supported by fellowships from the Kreitman School of Advanced Graduate Studies and Ben-Gurion University of the Negev, to which the authors are grateful. Author contributions Conceptualization: GM, SAMethodology: GM, SAInvestigation: GMVisualization: GMSupervision: SAWriting—original draft: GMWriting—review & editing: GM, SA. Funding This research is partly funded by the Israel Science Foundation (ISF) under grant No. 897/21 - “Deep learning for multidimensional classification of complex structured light beams in optical wireless communication applications”. Data availability All datasets, preprocessing scripts, and code used in this study are publicly available via the Zenodo repository (DOI: https://doi.org/10.5281/zenodo.18681790). Additional materials or clarifications can be provided upon reasonable request to the corresponding author, Gokul Manavalan ([email protected]). Declarations Competing interests The authors declare no competing interests. Footnotes Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. References 1. Qu, Z., Shi, W., Liu, B., Gupta, D. & Tiwari, P. IoMT-based smart healthcare detection system driven by quantum blockchain and quantum neural network. IEEE J. Biomed. 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(2025). 10.1145/3719276.3725200 Associated Data This section collects any data citations, data availability statements, or supplementary materials included in this article. Supplementary Materials Supplementary Material 1 (93.1KB, docx) Data Availability Statement All datasets, preprocessing scripts, and code used in this study are publicly available via the Zenodo repository (DOI: https://doi.org/10.5281/zenodo.18681790). Additional materials or clarifications can be provided upon reasonable request to the corresponding author, Gokul Manavalan ([email protected]). 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