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

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 qc.SE

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  • $\begingroup$ Can you give a mathematical condition? I..e, it would help if you can put your question into math... $\endgroup$
    – Tobias Fünke
    Commented Dec 10, 2023 at 8:36
  • $\begingroup$ Sure. What I mean is if the system is consisted of N subsystems so that the Hilbert space is partitioned as $H = H_1 + H_2 + ... + H_N$, then can we tell if $U_1 = U_2 A_1 A2 \dots$ where $A_1 = $, $A_2$, ... are operators acting only localy on each subspace $\endgroup$
    – george doultsinos
    Commented Dec 10, 2023 at 10:10
  • $\begingroup$ @georgedoultsinos Please edit the question accordingly. $\endgroup$
    – Norbert Schuch
    Commented Dec 10, 2023 at 15:28
  • $\begingroup$ My feeling is that the general problem is computationally hard (DQC1?). It is certainly a problem people thought about, so there should be information out there on the internet. ("LU equivalent"). Still, you need to specify (i) if this is for a fixed # of qubit (in which case it is probably not computationally hard), or a large number, and (ii) if the latter, how the gate is even specified (in that case, e.g., it could be a problem on the LU equivalence of two circuits.) $\endgroup$
    – Norbert Schuch
    Commented Dec 10, 2023 at 15:30

1 Answer 1

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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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