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?
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.