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

1

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?

what method can be applied to do what? – glS – 2020-11-27T19:09:34.420

It's mention in the question. An operator conditioned on the current state of system. If after doing some operations the current state of the system comes out to be $|\psi_0\rangle$ then I want to apply operator $U_1$ and so on. – Binshumesh sachan – 2020-11-28T03:58:48.873

you just stated that $U_i$ are applied to $|\psi_i\rangle$, that you start with $|\psi_{initial}\rangle$ and that you apply two unitary operations to it sending it to some $|\psi_i\rangle$ and that you then apply $U_i$ to this state. I see no question here. Also, it would be great if you could provide the context on why you want to do this – glS – 2020-11-28T10:25:14.990

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.