Are quantum operations reversible?

Question

If We perform some unitary operations on a Quantum State $|A\rangle$ after which it becomes$|A'\rangle$. Then if we perform the inverse of all those unitary operations on the state $|A'\rangle$ in reverse order, can we roll back to the state $|A\rangle$?

Answer 1

Any operation on quantum computer (with measurement being exception) are described by unitary matrix. A feature of the unitary matrix is $AA^\dagger=I$, which means that transpose conjugate to matrix $A$ is inverse to $A$ too. It can be easily proven that if $A$ is unitary then $A^\dagger$ is also unitary hence it is also quantum gate.

If you came from state $|\psi_0\rangle$ to $|\psi_1\rangle$ by transformation $A|\psi_0\rangle = |\psi_1\rangle$, then it is possible to reverse the transformation in this way: $A^{\dagger}|\psi_1\rangle = |\psi_0\rangle$.

In practise, this means that you put all gates in original circuit in reverse order and replace each gate $A$ with its transpose conjugate operator $A^{\dagger}$.