Let's say I have a universal quantum computer that can perform anything.
In this case, any VQE algorithm can use any operation, so no ansatz is needed to implement it.
My question is, is the usage of the ansatz for the VQE algorithms because current quantum computers lack the ability to perfrom the universal quantum computations within a given coherence time?
Understanding the term "ansatz" here to be that functional form which limits the trial state to some subset of the full Hilbert space spanned by the problem -
For many (most?) problems of interest, the Variational Quantum Eigensolver (VQE) does rely on this reduction in space in order to be tractable, even if you have constant-depth implementations of any unitary operator on any number of qubits.
To have an unrestricted functional form (which I would personally refer to as a "robust ansatz"), you would need to have a number of free parameters equal to the number of degrees of freedom in your Hilbert space.
For example, for the common case where we may want to access any pure state over $n$ digital qubits, this is equal to $2⋅2^n-2$ parameters. (There are $2^n$ elements in the statevector representation, and each one has a real part and an imaginary part. But we can remove one degree of freedom for the normalization constraint, and we can remove one more by fixing the global phase.) This is, notably, exponential in the number of qubits.
Even if we are guaranteed that our quantum computer can prepare any computational state at all with great speed and fidelity, we're still going to need to feed in all these parameters to tell it which state to prepare, and the optimization routine is still going to need to tell us how to update all these parameters. So the runtime for VQE using a robust ansatz spanning the whole computational Hilbert space is exponential.
Of course, many (most?) VQE problems do not need to explore the whole computational Hilbert space. For example, the common case in second-quantized chemistry problems with fixed particle number $η$ considers the much smaller Hilbert space with $2⋅{n\choose η}-2$ degrees of freedom, which scales asymptotically like $n^η$, exponential in particle number.
You can reduce this further with spin-conservation, and in principle you can exploit point-group symmetries in your system to reduce it even further. That is my personal favorite line of research with VQE. But I don't think it changes the asymptotics any (someone please correct me if I'm wrong!), so ultimately one must content oneself that robust VQE and scalable VQE don't mix.
The next best thing to hope for is a scalable ansatz which guarantees $ε$-close fidelity with the target state. I think the traditional "hardware efficient ansatz" achieves something to that effect, but I'm not well versed in that line of research. I'd appreciate if people could comment with relevant references.
If you indeed had a universal fault-tolerant quantum computer then you would not be using VQE, but instead go directly to Quantum Phase Estimation to find your solution.
