# Equivalence of two entangling operators with respect to local operators

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

• Can you give a mathematical condition? I..e, it would help if you can put your question into math... Commented Dec 10, 2023 at 8:36
• 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 Commented Dec 10, 2023 at 10:10
• @georgedoultsinos Please edit the question accordingly. Commented Dec 10, 2023 at 15:28
• 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.) Commented Dec 10, 2023 at 15:30

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.