Is there a way to rotate an unknown state towards another known state?

Question

I would like to rotate my $|\Psi\rangle$ state towards $|1\rangle$: $$ |\Psi\rangle= a|0\rangle + b|1\rangle \ \rightarrow \ |\Psi'\rangle= a'|0\rangle + b'|1\rangle$$ with $|a'| < |a|$, $|b'| > |b|$ and without prior knowledge of $a$, $b$.

I think a way to do it is to do a partial swap with an ancilla: $$ |\Psi\rangle|1\rangle_a \ \rightarrow \ |\Psi'\rangle|\phi\rangle_a $$

A criteria is that the strength of the rotation is parametrized $U(\theta)$. What happens to the ancilla $|\phi\rangle_a$ does not matter.

Is there something like a partial swap?

Thanks :)

Answer 1

$\newcommand{\1}{|1\rangle}$ $\newcommand{\0}{|0\rangle}$

seems to me like what you're looking for is a "reverse" amplitude damping channel.

Consider this circuit:

where the ancilla starts off at $\0$. If the initial state is $\1$, the first X gate will turn it into $|0\rangle$ and nothing will happen. If it's $\0$, the controlled Y will rotate the ancilla towards $\1$, which will then rotate the original qubit towards $|1\rangle$. The strength of the rotation will be determined by $t$.

If we calculate the action of this circuit on the data qubit $|\psi\rangle = a\0 + b\1$, the final state will be: $$ (a\cos\theta\0 +b\1)\0+(a\sin\theta\1)\1 $$ and so you can see that regardless of the state of the ancilla, the final amplitudes will be closer to $\1$.