Iterative methods are the dominant approach for solving large, sparse linear systems in scientific computing, particularly those arising from the discretization of partial differential equations, because direct solvers become impractical at scale. The performance of iterative solvers is governed primarily by the condition number of the system matrix, which controls convergence rates through its influence on the spectrum of the associated error-propagation operator. Consequently, a central objective of iterative linear algebra is the design algorithms that mitigate condition-number dependence and accelerate convergence. In this work, we introduce qRLS (quantum relaxation for linear systems), an iterative quantum framework for gate-based quantum computers that implements relaxation coherently in superposition. Classical relaxation methods rely on repeated application of a non-unitary contraction to progressively damp error modes, a mechanism that is fundamentally incompatible with gate-based quantum computation, where evolution must be unitary and reversible. In particular, the no-cloning theorem and the requirement of coherence across superposed states preclude naïve implementations of iterative schemes that depend on intermediate damping, discarding of iterates, or measurement-driven restart. The qRLS framework overcomes this challenge by embedding the relaxation process into a single larger quantum transformation, enabling the iterative evolution to be applied coherently without intermediate measurements. This construction preserves the structure and convergence intuition of classical relaxation while allowing amplitude amplification to be used to enhance solution preparation, thereby enabling systematic use of quantum resources within an iterative setting. We analyze the convergence behavior of qRLS for symmetric positive-definite linear systems, focusing on how the spectral properties of the relaxation operator determine contraction and condition-number scaling. This analysis identifies parameter regimes in which the quantum relaxation achieves near-optimal dependence on the condition number, matching the best achievable scaling for classical relaxation-based iterative solvers within this class. The resulting qRLS framework provides a modular and extensible pathway for translating classical relaxation-based linear solvers into the quantum domain, with direct relevance to large-scale problems in scientific and engineering computation.