  Superconducting quantum circuits are typically housed in conducting
enclosures in order to control their electromagnetic environment. As devices
grow in physical size, the electromagnetic modes of the enclosure come down in
frequency and can introduce unwanted long-range cross-talk between distant
elements of the enclosed circuit. Incorporating arrays of inductive shunts such
as through-substrate vias or machined pillars can suppress these effects by
raising these mode frequencies. Here, we derive simple, accurate models for the
modes of enclosures that incorporate such inductive-shunt arrays. We use these
models to predict that cavity-mediated inter-qubit couplings and drive-line
cross-talk are exponentially suppressed with distance for arbitrarily large
quantum circuits housed in such enclosures, indicating the promise of this
approach for quantum computing. We find good agreement with a finite-element
simulation of an example device containing more than 400 qubits.
