# Quantum gates for more than two basis states

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!

## 1 Answer

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.

• Thanks for taking the time to answer! I see, so I guess I could leave the target gate just as $U_{goal} = H_2$ and when it comes to calculating the infidelity I just cut out the top left $2\times 2$ part of my evolved gate... Thank you! Apr 22, 2022 at 16:05
• Yes, that's what I would do. Apr 25, 2022 at 6:32