Research On Quantum Algorithms Of Force Density Equations And Dense Matrix Gaussian Regression

After decades of development,cable-membrane structures are widely used in large buildings such as sports and entertainment facilities and public landscapes.In the design of the cable-membrane structure,the most critical problem is the shape determination of the cable-membrane structure,that is,the form-finding problem.Force-density form-finding is one of the most commonly used form-finding methods.The main work of this method is the solution of force density equations.However,with the increase of the number of discrete nodes and elements in the force density equation system,the classical algorithm is less efficient in solving the force density equation system.In this paper,the quantum algorithm is used to solve the force density equation system.Compared with the classical algorithm,the complexity is exponentially accelerated.The coefficient matrix of the force density equation system is a sparse matrix.For the more general dense matrix,the quantum Gaussian process regression of dense matrix is studied in this paper,which provides a new solution for using quantum computer to solve the Gaussian process prediction distribution problem.The main work of the paper is as follows:1.Deform the force density equations,use the quantum algorithm for solving linear equations(HHL algorithm)to solve the force density equations,and analyze the complexity of the algorithm.According to the positive definite symmetry of the coefficient matrix of the force density equation system,it is decomposed into the product of a sparse matrix and its own transposed matrix,and then the decomposed coefficient matrix is expanded into a higher-dimensional Hilbert space,and the HHL algorithm is used to find The method of sparse matrix inversion obtains the inverse of the coefficient matrix,and finally obtains the quantum state form corresponding to the solution of the force density equation system.It can be seen from the related complexity analysis that the above method achieves an exponential acceleration effect compared with the classical algorithm for solving the force density equation system.2.The quantum algorithm of Gaussian process regression of dense covariance matrix is studied.The dense covariance matrix is simulated using the Hamiltonian simulation method based on quantization and quantum signal processing,and the inverse of the covariance matrix is approximated by the inverse function of the covariance matrix.In the prediction stage of the process regression,a corresponding quantum algorithm is designed to calculate the mean and variance of the prediction distribution.The algorithm used in this paper relaxes the requirement on the sparsity of the covariance matrix and has faster computational efficiency than the classical algorithm.