There is something of a trend of a pendulum swinging between centralization and decentralization in computing.

Mainframes where you had to physically sit at and perform batch processing eventually had time-sharing capability added via remote terminals. Centralization came back when we got PCs, which then gave way to having data available on the internet. Our phones became computers, and then fast mobile data connections let us shift applications and processing into the cloud.

If it's physically possible to have a quantum computer at home, or in our pocket, we probably will. If I had to guess based on how we do things today, I'd say that those quantum processing capabilities will probably be used for co-processing for very specific applications. Maybe quantum cryptography? And anything more general, or requiring a large amount of qubits, will be available via the cloud.