Are quantum operations reversible?

Question

If We perform some unitary operations on a Quantum State $|Arangle$ 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 $|Arangle$?

Martin Vesely · Answer

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 $|psi_0rangle$ to $|psi_1rangle$ by transformation $A|psi_0rangle = |psi_1rangle$, then it is possible to reverse the transformation in this way: $A^{dagger}|psi_1rangle = |psi_0rangle$.
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 $A^{dagger}$.

Are quantum operations reversible?

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