How to apply a operator to qubit system on the basis of current state of system?

Question

Suppose I have three different operators $U_1, U_2,U_3$. Now, these three operators will be applied if my current state of the system is $|\psi_0\rangle,|\psi_1\rangle $ and $|\psi_2\rangle$ respectively.

Now suppose I started with some initial state $|\psi_{initial}\rangle$ and after applying two unitary operations it will be converted to one of the states above and on the basis of that the respective unitary operator needs to applied.

I know we can't measure the state as it will collapse the system. So, what method can be applied here?

Answer 1

I assume that you have a qubit register $q$ and given that the state of $q$ is $|\psi_i\rangle$ you want to apply $U_i$ to $q$ for $i=0,1,2$. If this is what you wish to do, then unfortunately if the states $|\psi_i\rangle$'s are not orthogonal to each other, then this kind of operations are not possible in a quantum setting in for any general $U_i$'s. This is not possible because such an operator is not unitary. For instance take the simple case of $q$ being in either the state $|0\rangle$ or $|+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}$ and I wish to apply $I$ on $q$ if it is in the state $|0\rangle$ and $H$ gate on $q$ if the state of $q$ is $|+\rangle$. This mathematically would mean that I need an operator $U$ that works as follows: $$U|0\rangle=|0\rangle \text{ and } U|+\rangle = \frac{1}{\sqrt{2}}(U|0\rangle + U|1\rangle) =|0\rangle.$$ It is quite obvious that $U$ is not reversible and hence is not a unitary. So such a quantum operation does not exist.

However, if the states $|\psi_i\rangle$'s are orthogonal to each other and the state in $q$ is one of these states and is not any superposition of these states, then certainly you can measure and then conditioned on the measured result you can apply the $U_i$ of your choice.