types of states that can be created with a given number of entangling gates

Question

I want to know if it is possible to say something in general about the "richness" or "complexity" of quantum states that can be created using a given number of entangling 2-qubit gates.

For example, using $n$ CNOTs, we can create an $n$ qubit GHZ state, and furthermore that we can't create it using less than $n$ such gates.

Trying to extend this logic, are there states that can only be created, for example, using no less than $O(n^2)$ 2-qubit entangling gates, and that are "more complex" than GHZ state by some suitable metric? In partcular, a metric that represents whether these states can represent for example a ground state of an interacting many body system, or something else that is of interest to compute?

Reason for asking

The motivation for asking this question is because I am interested in understanding how many qubits and entangling gates are needed to perform a useful quantum simulation. In this paper, fig. 1, they look at spin chains simulation that scales approximately as $n^2$ - I wanted to know how generic this is and if there is an underlying principle.

Answer 1

The question seems to ask for:

• A quantum state that requires $O(n^2)$ entangling gates to prepare, starting from some canonical state (say the all-0's ket)
• Wherein the state so-prepared is "of interest" to prepare with a quantum computer.

Considering initially the generalized GHZ/cat state $\frac{1}{\sqrt n}(|00\cdots 0\rangle+|11\cdots 1\rangle)$. As stated this can be prepared from the all-0's ket $|00\cdots 0\rangle$ with a Hadamard gate and $O(n)$ CNOT gates. Note, however, that the Hadamard and CNOT gates are from the Clifford gate set. By the Gottesman-Knill theorem, there likely will not be any computational speedup with a quantum computer preparing such states.

Consider also the generalized W state - $\frac{1}{\sqrt{n}}(|0\cdots 01\rangle + |0\cdots 010\cdots 0\rangle + |10\ldots 0\rangle)$. A circuit to prepare this state from the canonical all-0's ket does use non-Clifford gates such as arbitrary rotation gates. An initial circuit as described here naively uses $O(n^2)$ such gates. However, as explained in an answer from Craig Gidney, it appears that this can be reduced to $O(n)$ CSWAP gates. Nonetheless the CSWAP gate is not in the Clifford gate set, which seems to suggest that the generalized W state is "more complex" to prepare than preparation of the generalized GHZ state.

We can take this analysis a bit further and ask for a simple class of generalized states on $n$ qubits that (1) requires $O(n^2)$ gates to prepare, starting from some canonical state, wherein (2) these gates are not from the Clifford gate set? The GHZ state fails on both accounts, while the W state appears to fail only on the first requirement.

A slight variation of the W state may suffice - for example, the W state is in a superposition of "one-hot" basis states; perhaps a generalized "two-hot" state requires more than $O(n)$ non-Clifford gates (as there are $\frac n 2(n-1)$ such basis vectors).