In this work, the authors present a comparative analysis of quantum computers in the context of image processing. The benchmark was performed on seven quantum computers and one simulator. For the real quantum devices, three different families of QPU (quantum processing units) were used: ion-trap devices and superconducting qubits (the IBM Falcon and Eagle series). To encode 585 images on the quantum computers, the FRQI (Flexible Representation of Quantum Images) and LPIQE (Local Phase Image Quantum Encoding) methods were employed. The reconstructed images were compared using the MSE (Mean Squared Error) and PCC (Pearson Correlation Coefficient) metrics. The experimental protocol was implemented in Python, using the Qiskit, SciPy, and Geqie libraries. The results of the experiment are presented in tabular form, as histograms, and as a correlation matrix.This study investigated the correlation of MSE with (i) the QPU implementation technology (i.e., between ion-trap devices and superconducting qubits) and (ii) the type of superconducting quantum processor (Falcon vs. Eagle). Additionally, differences in the error distributions between these QPU series (Falcon vs. Eagle) were analyzed.The main conclusions are as follows: a strong correlation of MSE with the type of superconducting quantum processor (Falcon vs. Eagle) and a moderate correlation of MSE with QPU implementation technology (ion-trap vs. superconducting) were demonstrated. Furthermore, it was shown that Falcon-series processors exhibit a bimodal error distribution, whereas Eagle processors exhibit a unimodal distribution. No correlation was found between MSE and the error rate.

This study presents the Flexible Representation for Quantum Images (FRQI) Pairs method, a novel approach that leverages Quantum Recurrent Neural Networks (QRNN) for image classification. The proposed method achieves an accuracy of 74.60% on the full Modified National Institute of Standards and Technology (MNIST) handwritten digit data set, demonstrating its effectiveness in handling quantum encoded data for classification tasks.By reducing the size of the QRNN by the exponential factor, the FRQI Pairs method highlights the potential of integrating quantum computing principles with neural network architectures, offering a promising direction for advancing quantum machine learning.The research evaluates the FRQI Pairs method against existing quantum and classical models, demonstrating its competitive performance against other state-of-the-art approaches and showing potential for future advancements in the field. This research opens avenues for further exploration of quantum preprocessing and hybrid model architectures, marking a step forward in the application of quantum machine learning.

The current investigations present the numerical solutions of the novel singular nonlinear fifth-order (SNFO) system of multi-pantograph differential model (SMPDM), i.e., SNFO–SMPDM. The novel SNFO–SMPDM is obtained using the sense of the second kind of typical Emden–Fowler and prediction differential models. The features of shape factor, pantograph along with singular points are provided for all four obtained classes of the SNFO–SMPDM. The extensive use of the singular models is observed in the engineering and mathematical systems, e.g., inverse systems and viscoelasticity or creep systems. For the correctness of the proposed novel SNFO–SMPDM, one case of each class is numerically handled by applying supervised neural networks (SNNs) along with the optimization of Levenberg–Marquardt backpropagation scheme (LMBS), i.e., SNNs–LMBS. A dataset using the traditional variational iteration scheme is designed to compare the proposed results of each case of SNFO–SMPDM. The obtained approximate solutions of each class using the novel SNFO-SMPDM are presented based on the training (80%), authentication (10%) and testing (10%) measures to evaluate the mean square error. Fifteen numbers of neurons, and sigmoid activation function are used in this SNN process. To authenticate the competence, and precision of SNFO–SMPDM, the numerical simulations are accessible by applying the relative measures of regression, error histogram plots, and correlation.