Can a universal gate set be constructed using only $|0\rangle$ state prep, global arbitrary 1-qubit gates, local CZ gates, and global Z measurements?

Question

I'm exploring the possibilities of constructing a universal set of gates for quantum computation and was wondering about the following scenario:

Is it possible to implement a universal set of gates using only:

1. The ability to prepare all qubits in the $|0\rangle$ state.
2. Arbitrary single-qubit rotations that act globally (i.e., simultaneously on all qubits in the system).
3. Local controlled-Z (CZ) gates.
4. Global destructive measurements in the Z basis (simultaneously measuring all qubits in the computational basis).

I'm particularly interested in this question in the context of quantum computing with neutral atoms, where all-to-all connectivity can be achieved via atom movement.

If this is possible, how would one design the decomposition for an arbitrary quantum circuit in this setup? For example, could you provide an example of how to implement a local Hadamard gate on a specific qubit using these operations? If not, what specific limitations prevent universality with this gate set?

Any insights or references would be greatly appreciated!

Answer 1

Here's a Hadamard gate implemented with global single qubit gates and local CZ gates on a line. It requires a |1> and an |i> as catalysts:

Pretty inefficient, but it works.