Enhanced Variational Quantum Regression for Non-linear Curve Fitting in Higher Dimensions

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.