  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.
