Can the spin-1/2 720° rotation trick apply to qubits?

Question

The idea of a 4π rotation to return an electron to its original state instead of just a single 2π rotation exists - but can this idea apply to qubits or entangled qubits?

Are there any use cases for 360° or 720° quantum logic gates?

Answer 1

For a single qubit, the analogy is that 720° = $4\pi$ rotation is the unitary operator $I$, and 360° = $2\pi$ rotation is the unitary operator $-I$, while any other rotation is a unitary with $\text{det} = 1$, i.e. an element of the special unitary group $SU(2)$, which is isomorphic to the group of unit quaternions.

Even though $-I$ doesn't affect the state of a qubit, the controlled version Control-($-I$) is not the same as Control-($I$), which is still the identity.