Abstract: A generalization of the Harrow/Hassidim/Lloyd algorithm is developed by providing an alternative unitary for eigenphase estimation adopted from research in the area of quantum walks, which has the advantage of being well defined for any arbitrary matrix equation. The procedure is most useful for sparse matrix equations, as it allows for the inverse of a matrix to be applied with (Nnz log(N)) complexity, where N is the number of unknowns, and Nnz is the total number of nonzero elements in the system matrix. This efficiency is independent of the matrix structure, and hence the quantum procedure can outperform classical methods for many common system types. We show this using the example of sparse approximate inverse (SPAI) preconditioning, which involves the application of matrix inverses for matrices with Nnz=(N).