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My physics master thesis is about variational quantum optimization and I have to run algorithms with more qubits than 15. Is it possible to make any requests to use larger IBM real quantum computers?

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You can try to sign-up for the The IBM Quantum Researchers Program to have more flexibility to your account, but I don't think you will have access to their premium devices even with this option.

However, someone asked about this not too long ago here in this question and the answered by Lena is to use fake devices (simulators that are trying to mimic real hardware). The provided list of fake devices as pointed out by Lena's answer can be found here. So for instance, you can import in FakeToronto and use it to mimic the results you would get when running your circuit through imq_toronto. Of course, there will be differences but this at least give you some insight on what you might expect.

Hopefully this helps.

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I don't think there is any system in the free tier of IBM Q, the park of IBM's quantum systems, with more than $15$ qubits. The larger systems are reserved for 'partners', which in general are business that pay (a considerable amount at that).

You could try and request access to it, but as far as I am aware there is no paved road to do so, and you most likely will need some in-house contacts that you could ask. To be honest, I expect the chances of this approach succeeding to be slim, but don't let me stop you of course. If you find anyone, contact them preferably through your professor or some senior staff.

If the systems don't necessarily need to be from IBM, you could request access to Rigetti's system(s) via this link. It is not strictly speaking open access, but since you are a MSc. student I expect you to have a university email. Use that email and your chances of getting access are quite considerable, I would say.

You can program these machine(s) using python just as with qiskit <-> IBM Q, and their Aspen-8 QPU has $32$ qubits (in a rather interesting architecture, but I'm not sure what you need in that regard).

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