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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?

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    $\begingroup$ With 99% fidelity on your favorite representative gate (say, the square-root-of-SWAP), you'd be hard-pressed to get a depth of more than 1000 gates - which is what you might need, to take advantage of the all-to-all connectivity of the 1000 qubits that you have. $0.99^{1000}=0.000043$, pretty small. $\endgroup$
    – Mark Spinelli
    Aug 27, 2022 at 21:23
  • $\begingroup$ Maybe variational algorithms like VQE or QAOA would run successfully. $\endgroup$
    – Martin Vesely
    Aug 28, 2022 at 6:05
  • $\begingroup$ @MarkS nice argument! But why $n$ gates for $n$ qubits on full connectivity rather than, say, $n^2$? Also, is $0.5\times 10^{-4}$ really too small for, say, QPE? $\endgroup$
    – Nikita Nemkov
    Aug 28, 2022 at 10:45
  • $\begingroup$ Well, I think you need $n^2$ gates to have all qubits have an opportunity to talk with each other in a nearest-neighbor topology, by doing a bunch do SWAPS… Also, the overall fidelity may not be too small, you’re probably right about that. $\endgroup$
    – Mark Spinelli
    Aug 28, 2022 at 12:13
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    $\begingroup$ @MarkS well, $n$ gates on a linear topology is enough to produce a GHZ state, say. On the other hand, you need at least $n(n-1)/2$ gates to use all the potential that comes with the full connectivity. So my estimations are opposite to yours, yet they probably both make sense:) $\endgroup$
    – Nikita Nemkov
    Aug 28, 2022 at 12:42

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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.

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