Variational Quantum Eigen-Decomposition Preconditioning Method for Solving Computational Mechanics Problems

Recently, quantum computing emerged as a paradigm for solving systems of linear equations.However, large condition numbers significantly increase the computational complexity of quantum linear equation solvers.In this work, a new quantum preconditioning approach called variational quantum eigen-decomposition (VQED) is proposed, where the preconditioner is defined as a weighted projector onto the subspace spanned by the eigenvectors of a matrix.A variational quantum algorithm with projection-based deflation is performed to calculate eigenvalueeigenvector pairs for constructing the preconditioner.The proposed VQED method is used with quantum singular value transformation to solve linear systems for mechanics examples.It is demonstrated that the VQED can reduce the condition numbers to as low as 1.03, which is a significant improvement over the original condition number.