Exploration of Accelerated Solutions to Calculus Problems Based on Quantum Computing Algorithms

In large-scale scientific computing and engineering modeling, calculus problems are widely used as basic mathematical tools in physical simulation, control systems, image processing and other fields. However, their analytical and numerical solutions in high-dimensional and complex functions often face bottlenecks such as high computational complexity and slow convergence. To this end, this paper proposes a calculus acceleration solution method based on a quantum computing framework. First, this paper constructs a quantum state encoding mechanism suitable for function integration and differentiation operations. Then, Variational Quantum Algorithm (VQA) is applied to solve the optimal solution path in structured calculus expressions, and combined with Quantum Fourier Transform (QFT) to improve function boundary processing and analysis speed. Finally, a quantum-classical hybrid iterator is designed through a hybrid architecture with classical numerical methods. Experimental results show that in high-dimensional integration scenarios, compared with the classical Monte Carlo method, Quantum Amplitude Estimation (QAE) can reduce the number of queries from thousands to hundreds with the same error accuracy (such as); in the simulation experiment of ordinary differential equations, the simulated VQA shows a higher convergence efficiency with the same number of iterations. The above results fully demonstrate the acceleration potential and solution accuracy advantages of quantum algorithms in high dimensions and complex boundary conditions, and provide new ideas for the application of calculus problems in large-scale scientific and engineering calculations.