In what sense is it 'more complicated to navigate through a Hilbert space to find more complex states'?

Question

In the lecture titled Entanglement and Complexity: Gravity and Quantum Mechanics, Professor Leonard Susskind implies (at time 17:49) that the complexity of a quantum state impacts how 'complicated' it is to 'navigate your way through the Hilbert space to find that state'.

I've heard this and similar sentiments a few times. Why is it true? How are Hilbert spaces 'navigated through' and why does seeking a more 'complex' [*] state make this navigation more 'difficult' [**]?

[*] meaning more entangled?

[**] difficult in what sense? Time consuming?

Answer 1

One navigates through Hilbert space (and indeed through life) by unitary transformations, since time evolution is always unitary. Furthermore, since the Hamiltonian only allows local interactions, these unitary operators are always local unitary operators.

By some kind of definition, the complexity of a quantum state is how many local unitary operators one needs to use to get close to it from some reference state.

In lattice quantum systems we do this by counting the number of quantum gates one needs to get to the target state from a product state. It's very hard to compute this number.

In continuum quantum systems, probably the next best thing is to measure the integrated norm of the Hamiltonian $H(t)$ along a path. This is a lot like action, hence complexity = action.

"Difficulty" is measured by Hamilton's principle, which says that the classical universe extremizes the action. A "difficult path" is one whose action is far from extremized.