2
$\begingroup$

Consider a fixed set of gates of the kind $\land_n(P_{\varphi})$. With $P_{\varphi}$ being a relative phase shift gate by $\varphi$.

Can I assert that any permutation of that set is equivalent?

Example:

enter image description here

Can I rearrange this circuit at my liking and always get the same evolution?

$\endgroup$

1 Answer 1

5
$\begingroup$

Yes, as long as you change the order of the gates without changing the qubits on which each gate acts.

Proof: All phase gates are represented by diagonal unitary matrices. If $A$ has a diagonal matrix then controlled-$A$ also has a diagonal matrix. Therefore, all gates in your set of gates have diagonal matrices. Conclusion follows from the fact that diagonal matrices commute.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.