Comparison of Classical and Quantum-Inspired Root-Finding Schemes on the Polynomial Equation x⁷ – 5x² – 108 = 0

Abstract

This paper investigates four iterative root-finding schemes for the nonlinear scalar equation x⁷ - 5x² - 108 = 0. We consider (a) the classical Newton–Raphson method, (b) a third-order MITAT-ROOT method based on a circle–tangent interpretation, (c) a quantum-inspired Newton–Raphson method, and (d) a quantum-inspired MITAT-ROOT method. The quantum-inspired variants introduce a “probability amplitude” that adaptively scales the iteration step in analogy with quantum amplitudes and phase evolution. All four methods are derived and discussed in detail, and their performance is compared in terms of convergence speed, iteration count, and execution time. Numerical experiments show that MITAT-ROOT and its quantum-inspired variant converge in fewer iterations than the classical Newton–Raphson method, at the cost of slightly more arithmetic per iteration due to the use of second derivatives and amplitude updates. A Python implementation with colorful graphical output is provided to visualize the convergence paths of all four methods on the given polynomial.