This question seems slightly naive. The Hilbert pace of any 2-level quantum system is given by the Bloch sphere and the algebra of observables arises from $SU(2)$, the Lie group generated by the three Pauli matrices $\sigma_x,\sigma_y,\sigma_z$.
When considering spin chains, relevant for example in the NISQ quantum devicesand for algrithms such as the VQAs, the corresponding state space as far as I understand is $\mathcal{H}_{m} = SU(2) \times \ldots \times SU(2)$, that is for a length $N$ spin chain, $N$ copies of $SU(2)$.
If my understanding is correct, I struggle to see why many people claim that a $N$ qubit system's Hilbert space is described by $SU(2^N)$. Naturally, this is quite differnt to $\mathcal{H}_N$ described before, not only as a manifold, but also in terms of its generators which are definitely not matching the dimensions of a 2-level system for $N>1$.
For example, in this talk by a very senior researcher (at 0:11:00).
Therefore my questions boils down to:
- Is there some map between $f: \times_N SU(2) \to SU(2^N)$ for $N>1$?
- If not, isn't the claim that the Hilbert space of $N$ qubits is $SU(2^N)$ false?