A technique relates a superconducting microwave cavity. An array of posts has different heights in the cavity, and the array supports a localized microwave mode. The array of posts includes lower resonant frequency posts and higher resonant frequency posts. The higher resonant frequency posts are arranged around the lower resonant frequency posts. A first plate is opposite a second plate in the cavity. One end of the lower resonant frequency posts is positioned on the second plate so as to be electrically connected to the second plate. Another end of the lower resonant frequency posts in the array is open so as not to form an electrical connection to the first plate. Qubits are connected to the lower resonant frequency posts in the array of posts, such that each of the qubits is physically connected to one or two of the lower resonant frequency posts in the array of posts.