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Many current devices are constrained to nearest neighbor connectivity or small system sizes, but suppose that a NISQ machine with 99-99.5% level two-qubit gate fidelities and arbitrary connectivity were available, with 1000 qubits. Are there really interesting algorithms that could be run that are not otherwise accessible on a similar-sized nearest-neighbor connected machine?

As an example, are there explicit quantum chemistry problems where this all-to-all connectivity would be a huge advantage?

shixian105
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The first thing I would do is ask the question: how good are they?

The comments to your question have already hinted at the answer. Not so good.

All-to-all connectivity is a dream for some platforms (e.g. superconducting) and very hard to realise on others (e.g. trapped ions). In fact, I would be happy to have an all-to-one to start with, given that it is a good CNOT connection. Note that the error rates of CNOT gates are much higher than Rz rotations.

With a thousand qubits and say a constant depth of five, I would try a very simple problem of finding the ground state energy of a mean-field model. The answer would be known and it would serve as a benchmark (i.e. can your QC find it?). Such a simple problem is not meant to show any 'advantage' which remains arguable for the 2019 supremacy paper too. Just don't ask what 'advantage' means.

I would return your 1000 qubits and tell you that I would be happier if you gave me 100 qubits with 1000 low-error-depth. On such a device I would run VQE with a highly customised (variational Hamiltonian) ansatz to find the ground state energy of the Hubbard model at any filling.

aimedaca
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