The advent of quantum computing, characterized by limited resources and qubits, the large circuit depth presents a substantial challenge for quantum algorithms designed to solve linear systems of equations. This paper proposes an innovative approach, named Enhanced Variational Quantum Regression (EVQR), specifically designed for solving non-linear curve fitting problems in higher dimensions and orders. It is a hybrid quantum-classical algorithm which not only surpasses the limitations of traditional one-dimensional solvers but also leverages variational principles to address complex problems. Our methodology centers around an initial solution known as an “ansatz” parameterized by a function. By employing quantum circuits, qubits are prepared to efficiently compute inner products in the quantum domain, facilitating accelerated and powerful calculations. The estimated inner product is then transmitted back to the classical computer, where classical optimization techniques such as Levenberg-Marquardt are employed to estimate the function parameters. This iterative process is repeated until the estimation reaches a desired level of accuracy within a predefined tolerance. Notably, we empirically observe that the time complexity of EVQR scales efficiently with the system size N. By leveraging the capabilities of SciPy's curve-fit and the robust optimization techniques of the Levenberg-Marquardt algorithm, our approach overcomes the challenge of higher-dimensional curve fitting. Our primary contribution lies in the fusion of quantum and classical methodologies, a novel integration that, unlike previous quantum approaches, outperforms in terms of performance and accuracy. By combining the strengths of classical optimization and quantum computations, EVQR offers a versatile framework for solving challenging problems, opening new possibilities for data analysis & modeling.
The advent of quantum computing, characterized by limited resources and qubits, the large circuit depth presents a substantial challenge for quantum algorithms designed to solve linear systems of equations. This paper proposes an innovative approach, named Enhanced Variational Quantum Regression (EVQR), specifically designed for solving non-linear curve fitting problems in higher dimensions and orders. It is a hybrid quantum-classical algorithm which not only surpasses the limitations of traditional one-dimensional solvers but also leverages variational principles to address complex problems. Our methodology centers around an initial solution known as an “ansatz” parameterized by a function. By employing quantum circuits, qubits are prepared to efficiently compute inner products in the quantum domain, facilitating accelerated and powerful calculations. The estimated inner product is then transmitted back to the classical computer, where classical optimization techniques such as Levenberg-Marquardt are employed to estimate the function parameters. This iterative process is repeated until the estimation reaches a desired level of accuracy within a predefined tolerance. Notably, we empirically observe that the time complexity of EVQR scales efficiently with the system size N. By leveraging the capabilities of SciPy's curve-fit and the robust optimization techniques of the Levenberg-Marquardt algorithm, our approach overcomes the challenge of higher-dimensional curve fitting. Our primary contribution lies in the fusion of quantum and classical methodologies, a novel integration that, unlike previous quantum approaches, outperforms in terms of performance and accuracy. By combining the strengths of classical optimization and quantum computations, EVQR offers a versatile framework for solving challenging problems, opening new possibilities for data analysis & modeling.
Abstract Variational quantum algorithms are promising methods with the greatest potential to achieve quantum advantage, widely employed in the era of noisy intermediate-scale quantum computing. This study presents an advanced variational hybrid algorithm (EVQLSE) that leverages both quantum and classical computing paradigms to address the solution of linear equation systems. Initially, an innovative loss function is proposed, drawing inspiration from the similarity measure between two quantum states. This function exhibits a substantial improvement in computational complexity when benchmarked against the variational quantum linear solver. Subsequently, a specialized parameterized quantum circuit structure is presented for small-scale linear systems, which exhibits powerful expressive capabilities. Through rigorous numerical analysis, the expressiveness of this circuit structure is quantitatively assessed using a variational quantum regression algorithm, and it obtained the best score compared to the others. Moreover, the expansion in system size is accompanied by an increase in the number of parameters, placing considerable strain on the training process for the algorithm. To address this challenge, an optimization strategy known as quantum parameter sharing is introduced, which proficiently minimizes parameter volume while adhering to exacting precision standards. Finally, EVQLSE is successfully implemented on a quantum computing platform provided by IBM for the resolution of large-scale problems characterized by a dimensionality of 220.
Quantum computing and artificial intelligence (AI) are positioned as groundbreaking tools capable of tackling complexities inherent to modern computation posed by high-dimensional optimization problems.This research presents a quantum-enhanced optimization framework that leverages the QAOA (Quantum Approximate Optimization Algorithm) and VQE (Variational Quantum Eigen solver) within a hybrid quantum-classical paradigm.Machine learning models, including reinforcement learning and NN (neural networks), are integrated to optimize the quantum circuit parameters, enhance convergence, and improve scalability for complex problem spaces.Experimental evaluations demonstrate that this approach outperforms traditional optimization techniques in terms of efficiency and solution accuracy, particularly for large-scale multi-objective tasks.The framework is applied to key domains such as smart energy systems, logistics, and scientific modeling, illustrating its versatility and alignment with sustainable development goals.This study also explores the limitations of current quantum hardware and discusses future pathways for advancing hybrid quantum AI systems, offering significant contributions to next generation computational methodologies and intelligent problem solving.
Gaussian processes are widely known for their ability to provide probabilistic predictions in supervised machine learning models. Their non-parametric nature and flexibility make them particularly effective for regression tasks. However, training a Gaussian process model using standard methods requires matrix inversions with a cubic time complexity, which poses significant computational challenges for inference on larger datasets. Quantum algorithms, such as the HHL algorithm, have been proposed as solutions that overcome the need for classical matrix inversions by efficiently solving linear systems of equations using quantum computers. However, to gain a computational advantage over classical algorithms, these algorithms require fault-tolerant quantum computers with a large number of qubits, which are not yet available. The Variational Quantum Linear Solver is a hybrid quantum-classical algorithm that solves linear systems of equations by optimizing the parameters of a variational quantum circuit using a classical computer. This method is especially suitable for noisy intermediate-scale quantum computers, as it does not require many qubits. It can be used to compute the posterior distribution of a Gaussian process by reformulating the matrix inversion into a set of linear systems of equations. We empirically demonstrate that using the Variational Quantum Linear Solver to perform inference for Gaussian process regression delivers regression quality comparable to that of classical methods.
Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. Here, we propose a hybrid quantum-classical algorithm, called Variational Quantum Linear Solver (VQLS), for solving linear systems on near-term quantum computers. VQLS seeks to variationally prepare | x ⟩ such that A | x ⟩ ∝ | b ⟩ . We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision ϵ is achieved. Specifically, we prove that C ⩾ ϵ 2 / κ 2 , where C is the VQLS cost function and κ is the condition number of A . We present efficient quantum circuits to estimate C , while providing evidence for the classical hardness of its estimation. Using Rigetti's quantum computer, we successfully implement VQLS up to a problem size of 1024 × 1024 . Finally, we numerically solve non-trivial problems of size up to 2 50 × 2 50 . For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in ϵ , κ , and the system size N .
The potential of quantum computing for scientific and industrial breakthroughs is immense, however, we are still in the Noisy Intermediate-Scale Quantum (NISQ) era, where the currently available quantum devices contain small numbers of qubits, are very sensitive to environmental conditions and prone to quantum decoherence. Even so, existing NISQ computers have already been shown to outperform conventional computers on specific problems and the key question is how to make use of today’s NISQ devices to achieve quantum advantage in the field of computational science and engineering (CSE). In this direction, this work proposes a hybrid computing formulation by combining quantum computing with machine learning for accelerating the solution of parameterized linear systems in NISQ devices. In particular, it focuses on the Variational Quantum Linear Solver (VQLS), which is a hybrid quantum- classical algorithm for solving linear systems of equations. VQLS employs a short-depth quantum circuit to efficiently evaluate a cost function related to the system solution. The algorithm’s circuit consists of a quantum gate sequence (unitary operators) that involves a set of tunable parameters. Then, well-established classical optimizers are utilized to tune the parameters of the sequence in order to minimize the cost function, which is equivalent to finding the system solution at an acceptable level of accuracy. In this work, it is demonstrated that machine learning tools such as feed-forward neural networks and nearest-neighbor interpolation techniques can be successfully employed to accelerate the convergence of the VQLS algorithm towards the optimal values for the circuit parameters when applied to parameterized linear systems that need to be solved for multiple parameter instances.
This research investigates a quantum enhanced cloud intelligence framework based on hybrid variational models that integrate Variational Quantum Algorithms (VQAs), Parameterized Quantum Circuits (PQCs), and classical machine learning optimization. The study aims to address the scalability and computational limitations of conventional cloud-based machine learning by leveraging the expressive power of quantum feature spaces and entanglement-driven representations. A structured methodology is presented, encompassing hybrid model design, dataset preparation, quantum circuit construction, and the implementation of a cloud-integrated training loop. Performance benchmarking across high-dimensional datasets demonstrates that the proposed hybrid approach can achieve faster training, improved model accuracy, and enhanced energy efficiency compared to classical baselines. The research further outlines the practical challenges posed by NISQ-era hardware, noise sensitivity, cloud latency, and hybrid optimization instability. Despite these limitations, the findings reveal strong potential for deploying quantum-assisted intelligence in real-time analytics, complex optimization problems, healthcare diagnostics, autonomous systems, and cybersecurity applications. This study contributes a unified integration framework, novel empirical benchmarks, and a practical roadmap for advancing quantum cloud synergy, positioning hybrid variational systems as a promising foundation for next-generation scalable machine learning.