How does the CNOT gate operate when the control qubit is a superposition?

Question

If a control qubit is in superposition, how it will affect target qubit if it is collapsed or in superposition? Is it true that CNOT works only if the control bit collapsed to 1? Also, is it possible to collapse or Hadamard control qubit “on the go” in a real life quantum computer and have a functional CNOT gate?

Answer 1

By an example with a control qubit in superposition and the target in $ |0\rangle $ state:

$$ \frac{|0\rangle + |1\rangle}{\sqrt{2}} |0\rangle = \frac{|0\rangle|0\rangle + |1\rangle |0\rangle}{\sqrt{2}}$$

Applying a CNOT will have the following result: $$ \frac{ CNOT(|0\rangle|0\rangle + |1\rangle |0\rangle)}{\sqrt{2}} = \frac{ CNOT(|0\rangle|0\rangle) + CNOT(|1\rangle |0\rangle)}{\sqrt{2}} = \frac{ |0\rangle|0\rangle + |1\rangle |1\rangle}{\sqrt{2}}$$

That is the CNOT acts linearly with a control qubit in superposition, but will change the target only on the part involving a $ |1\rangle$ in the control qubit.