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.
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