Quantum Iterative Algorithm for Linear Systems of Equation

We propose a Quantum Iterative Algorithm as a solution to the linear system of equations subject to certain constraints. The existing Quantum Solution of the linear system of equations of the form Ax=b, where A is a Hermitian Matrix, has been developed by Aram Harrow, Avinatan Hassidam and Seth Loyd. Solving a system of linear equations arises in many complex scientific problems. Here, one can find an approximate solution where A is a sparse square N-dimensional matrix with condition number K. The quantum algorithm, as proposed, has run time poly(LogN,K) compared to the classical case where it takes \(O(N \sqrt{\kappa })\) time. But the main problem in the NISQ era is the requirement of many qubits for solving such a linear equation system in the quantum domain. The algorithm proposed here requires very few qubits because of its iterative characteristics, which are runnable in modern-day available quantum machines compared to the HHL Algorithm.