Preconditioning for a Variational Quantum Linear Solver

We apply preconditioning, which is widely used in classical solvers for linear systems $A\mathbf{\text{x}}=\mathbf{\text{b}}$, to the variational quantum linear solver. By utilizing incomplete LU factorization as a preconditioner for linear equations formed by $128\times 128$ random sparse matrices, we numerically demonstrate a notable reduction in the required ansatz depth, demonstrating that preconditioning is useful for quantum algorithms. This reduction in circuit depth is crucial to improving the efficiency and accuracy of Noisy Intermediate-Scale Quantum (NISQ) algorithms. Our findings suggest that combining classical computing techniques, such as preconditioning, with quantum algorithms can significantly enhance the performance of NISQ algorithms.