# How does $|00\rangle$ evolve through an Hadamard and a CNOT gate?

If we have this given circuit:

So the output for $$|0\rangle$$ will be: $$\frac{1}{\sqrt{2}}\left(|00\rangle + |11\rangle\right)$$
And we have this given circuit:

What will be the output for $$|00\rangle$$ and why?

The Hadamard gate is its own inverse. Hence, the two Hadamards on the first qubit cancel, leaving the control qubit in state $$|0 \rangle$$. The bottom Hadamard gate will put the target qubit in the state $$\frac{1}{\sqrt{2}} (|0\rangle + |1\rangle)$$. Since the control qubit is still in state $$|0\rangle$$, the CNOT will do nothing to the target. Your output state is therefore $$\frac{1}{\sqrt{2}} (|00\rangle + |01\rangle)$$.