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 $|A\rangle$?

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    $\begingroup$ 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? $\endgroup$ Jun 25, 2020 at 22:43

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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 $A^{\dagger}$.

  • $\begingroup$ "Any operation on quantum computer (with measurement being exception) are described by unitary matrix". This is only true for pure states. The most general formulation of quantum dynamics is via CP-maps (with the states described by density matrices) and the only invertible CP maps (assuming input-output dimensions are the same) are unitaries; namely, every non-unitary CP map is not invertible. So, I would at least append the original statement. $\endgroup$ Jun 28, 2020 at 0:40
  • $\begingroup$ @keisuke.akira: Thanks for the comment, however, I am a little bit confused. I said that only operation described be unitaries are reversible, others are not. The others are measurement which cannot be reversed. I seems to me that you comment says the same. Or what am I missing? $\endgroup$ Jun 28, 2020 at 6:23
  • $\begingroup$ My disagreement is with the first sentence of your answer, "Any operation on quantum computer (with measurement being exception) are described by unitary matrix". This seems a bit ambiguous: I think a more precise statement would be to specify that CP maps encompass both unitaries and beyond, and are generally non-invertible -- even if we don't think of them as a (generalized) measurement. $\endgroup$ Jun 28, 2020 at 9:04
  • $\begingroup$ @keisuke.akira: OK, sure. I have to say that I see quantum computing mainly from programmer (or engineer) point of view. As far as I know, any quantum gate used in gate-based models should be reversibl, however, I see that you have deeper understanding of general models of QC than I. Please feel free to edit my answer in order to be more accurate. $\endgroup$ Jun 29, 2020 at 7:08
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    $\begingroup$ Your answer is correct @MartinVesely but what keisuke is trying to say is measurement can't be reversed. $\endgroup$ Aug 8, 2020 at 14:32

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