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?


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


1 Answer 1


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.


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