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Suppose in "near future" that we have quantum computers with thousands of qubits that can run Grover, HHL, and so on. Then, can we hope in the "near future" to have some web application that will run quantum algorithms in real-time?

For example, suppose Google or IBM with their quantum computers have a web application. A client sends a request on this application, therefore a classical computer analyses the request and detects an inversion of a huge matrix (that satisfied the HHL conditions). Then the classical computer sends the matrix to the quantum computer that solves instantly the problem with the HHL algorithms and returns the result to the classical computer. Finally, the classical computer sends back the result to the client. All this in a few milliseconds.

Is this example possible or will we have only applications like on the IBMQ that take some hours to be handled?

1Yes, your example is pretty realistic. In fact, that would be the best solution as quantum processors are useful only for a very narrow range of problems. – Sanchayan Dutta – 2019-10-21T13:09:57.350

1That's pretty much the direction most companies seem to be taking atm. Your last paragraph is confusing though. IBMQ is one example of this kind of thing: you send requests via the web, the computation is done on the quantum device, and you are given the results back. The fact that it's "slow" atm is just due to practical limitation of the current devices, the number of people using it, etc. – glS – 2019-10-22T10:40:07.530