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Looking at the single-qubit, toy example of VQE it's pretty much trivial that arbitrary X and Y rotations are sufficient to cover all of state space for our toy system.

Unfortunately, the toy example doesn't do enough to illustrate to me why it would be any harder to do the same for a larger system. So in what situations is it hard to find an ansatz?

EDIT

After reading through some of the resources from the accepted answer I came up with a response that works best for me:

  • The number of parameters required to describe all possible states of an n-qubit quantum system scales exponentially with n.
  • So an ansatz which can cover all possible states would need an exponential number of parameters. And that just won't do because:

    • We'd then need a classical optimistion algorithm which can search through an exponentially large parameter space.
    • We'd need an exponential number of gates to actually prepare the state (certainly not good for a NISQ device with short coherence times).
  • So we actually need to be able to find ansatzes who's parameters grow at most polynomially with the size of the system. But then of course, we can't cover all states.

  • So then the challenge is in balancing the tradeoff between keeping the number of ansatz parameters small, but still being confident that the spanning space of the ansatz covers our ground state.

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    $\begingroup$ Based on the direction you're headed in your edit, you might be interested in looking into tensor networks. As you noted, Hilbert space is huge (exponentially so), but physically relevant regions of Hilbert space are constrained by locality. This paper is a nice introduction, section 3.4 in particular is directly relevant to your comments above. $\endgroup$ – Jonathan Trousdale Jun 4 at 15:11
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    $\begingroup$ Thanks a lot! I've been coming back to your links one by one as I've been working through this all. (will delete this comment shortly as I believe it's not correct SEtiquette) $\endgroup$ – Alexander Soare Jun 5 at 7:00
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If you haven't read the Qiskit chapter on Simulating Molecules Using VQE yet, that's a good place to start. There's also a related response to a similar question here, which you might find helpful.

If you want to see an example of a problem that researchers are actively dealing with, you might try reading up on the ongoing progress around FeMoco simulation. A better understanding of this molecule could drastically reduce the energy cost of fertilizer production (which is about 1.2% of worldwide energy consumption), so it's a very attractive target for quantum computing.

This paper from Google was a significant advance in that effort, and it has a good introduction section that should give you a sense of the challenges that researchers are facing in constructing effective and efficient molecule simulations.

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    $\begingroup$ Wow! Thank you. There are probably many ways to answer this question but you somehow nailed it for me with the practical approach $\endgroup$ – Alexander Soare May 25 at 19:26

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