Lets say I have a time dependent hamiltonian of the form $$H(t) = H_0 + f(t) H_{drive}$$ (where $H_0$ may represent the hamiltonian for a superconducting qubit) and I want to find the optimal pulse $f(t)$, such that the infidelity between some target gate $U_{goal}$ and evolved gate $U(T)$ is minimal. Where $$U(T) = {\mathcal T}\exp\left(\int_0^T -iH(t)dt\right)$$
If I where to include the first $d$ basis states of the qubit into my optimization, the resulting hamiltonian, and therefore also the evolved gate $U(T)$, would be of dimension $d\times d$.
The question: How would my target gate $U_{goal}$ look like for cases with $d>2$? Because, for example, if my target gate would be the standard Hadamard gate (which has dimensions of $2\times 2$), the dimension of $U_{goal}$ and $U(T)$ won't be identical anymore.
Would it be of the form $$U_{goal} = H_2 \oplus \mathbf{I}_{}$$ (example for $d=4$)
$$\begin{bmatrix}1/\sqrt{2} & 1/\sqrt{2} & 0 & 0\\1/\sqrt{2} & -1/\sqrt{2} & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1 \end{bmatrix}$$
or would it be something else?
I would highly appreciate any input on this, thank you!
