# Are quantum operations reversible?

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 $$|A\rangle$$?

• What you've described is the reversibility of unitary transformations and not the "reversibility of the state". Unitaries are invertible maps -- in fact, the set of all unitaries forms a group -- and so yes, you will revert back to the original state. I'm not sure what you mean by the state being "reversible"? Do you mean invertible as a matrix? – keisuke.akira Jun 25 at 22:43

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 $$A^{\dagger}$$.