Quantum computing is a rapidly developing technology with great potential for scientific and engineering computation applications. Gate-based quantum computers provide exponentially large linear spaces and can potentially exponentially speed up the numerical solution of partial differential equations. Current work in gate-based quantum computing has not explored iterative methods for the solution of partial differential equations. Although non-iterative solvers for the quantum linear system problem can provide exponential speedups, their dependence on the condition number can eliminate potential speedups [1]. Here, we explore the first iterative approach for solving linear systems and partial differential equations on gate-based quantum computers. Based on the evolution of a first-order system of ordinary differential equations, a novel ODE relaxation method is shown to be strictly convergent for positive definite systems with no damping. Optimal damping for ODE relaxation is derived, and faster convergence is shown compared to optimally damped Jacobi relaxation [2]. The relaxation scheme is block-encoded in a linear system, whose solution provides the output of the iterations and is exponentially efficient on gate-based quantum computers compared to classical methods. The quantum multigrid method (qMG) is realized by applying the ODE relaxation method in a multigrid V-cycle on gate-based quantum computers. The linear operations of the multigrid algorithm are encoded as a block-encoded linear system, whose solution provides the outputs of the multigrid operations along with the output of a V-cycle. A detailed analysis of the blockencoded linear system investigating its condition number and the success probability of extracting the output is presented. Although solving the system is exponentially efficient, extracting the desired output vector from the block solution is expensive due to inter-grid transfer operations, limiting the application of subsequent V-cycles. Numerical results of the ODE relaxation and qMG methods are presented to demonstrate their convergence on example problems.