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Suppose that $U_1$ and $U_2$ are two (entangling) operators that act on a quantum system consisting of several subsystems. Is there any criterion to tell if these two are equivalent up to applying operators acting only locally to the subsystem?

For example the 2-qubit Control-Z phase gate can be transform to the Control-NOT via applying (local) Hadamard gates to the second (target) qubit before and after. However this is trivial. How one can tell in more complicated cases?


Cross-posted on physics.SE

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2 Answers 2

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Equivalent $U_1$ and $U_2$ should have the same spectrum. If eigenvalues of $U_1$ are all different, then there is essentially a unique unitary $V$ (up to a set of phases) which gives the equivalence $VU_1V^\dagger = U_2$. If $U_1,U_2$ are written in the eigenbasis, then matrices $V' = DV$, where $D$ is any diagonal unitary, form the set of all unitaries that give the equivalence between $U_1$ and $U_2$. We then can check if $V' = V_1 \otimes V_2$ for some $D$, which should be easy.

On the other hand, I don't think there is an efficient algorithm in a general situation. Let $S_1$, $S_2$ be eigensubspaces of $U_1$, $U_2$ that correspond to some eigenvalue $\lambda$. Clearly, $\dim S_1 = \dim S_2$. If $U_1$ and $U_2$ are equivalent by some $V_1 \otimes V_2$, then the minimum (and maximum) Schmidt rank of a vector in $S_1$ and in $S_2$ should coincide. In general, finding Schmidt rank of a mixed state is a hard problem. The separability problem is simpler, but it's known that it's NP-hard. I don't know if it's proven that finding Schmidt rank of a projector $P$ is also NP-hard, but I haven't seen efficient solutions of this problem either.

Edit

I assumed that $U_1$ and $U_2$ are locally equivalent if $$ (V_1 \otimes V_2)U_1(V_1 \otimes V_2)^\dagger = U_2 $$ for some local unitaries $V_1,V_2$. More broadly, we can define local equivalence by $$ (V_1 \otimes V_2)U_1(V_3 \otimes V_4) = U_2. $$ Under this definition the problem looks even harder, I think.

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Even checking if a quantum circuit is equal to the identity circuit is already QMA-hard. Thus, the general problem on $N$ qubits, provided that the unitary is specified by a circuit, is a computationally hard problem.

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