We introduce a simplification algorithm for meshes composed of quadrilateral elements. It is reminiscent of edge-collapse based methods for triangle meshes, but takes a novel approach to the challenging problem of maintaining the quadrilateral connectivity during level-of-detail creation. The method consists of a set of unit operations applied to the dual of the mesh, each designed to improve mesh structure and maintain topological genus. Geometric shape is maintained by an extension of a quadric error metric to quad meshes. The technique is straightforward to implement and efficient enough to be applied to real-world models. Our technique can handle models with sharp features, and can be used to re-mesh general polygonal, i.e. tri- and quad-dominant, meshes into quadonly meshes.