Quantum gates for more than two basis states

Question

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!

Answer 1

If you want the best possible evolution for your qubit space, then you simply don't define what your target is on the rest of the space - so long as you always start in the two-dimensional subspace/subsystem and end there, it doesn't matter what happens outside it. So, you'd probably set $$ U_{goal}=H_2\oplus U' $$ where $U'$ is any (unspecified) unitary. Your measure of the infidelity should only be over the two-dimensional component.