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Quantum linear system algorithms (QLSAs) for gate-based quantum computing can provide exponential speedups for solving linear systems but face challenges when applied to finite element problems due to the growth of the condition number with problem size. Furthermore, QLSAs cannot use an approximate solution or initial guess to output an improved solution. Here, we present Quantum Relaxation for Linear System (qRLS), as an iterative approach for gate-based quantum computers by embedding linear stationary iterations into a larger block linear system. The condition number of the block linear system scales linearly with the number of iterations independent of the size and condition number of the original system. The well-conditioned system enables a practical iterative solution of finite element problems using the state-of-the-art Quantum Signal Processing (QSP) variant of QLSAs, for which we provide numerical results using a quantum computer simulator. The iteration complexity demonstrates favorable scaling relative to classical architectures, as the solution time is independent of system size and requires O(log(N)) qubits. This represents an exponential efficiency gain, offering a new approach for iterative finite element problem-solving on quantum hardware. |
