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I am working on the classical simulation of quantum circuits. I know how to efficiently implement the following entangling gate, which -- in the following paper: https://arxiv.org/pdf/1803.02118 -- was declared to be an entangling gate that is sufficient to realize arbitrary quantum circuits together with 1-qubit gates. The gate is given by

$$ U_\text{entangle}=\frac{1}{2}\begin{pmatrix} ie^{i\pi/4} & e^{i\pi/4} & e^{i\pi/4} & -ie^{i\pi/4} \\ e^{-i\pi/4} & ie^{-i\pi/4} & -ie^{-i\pi/4} & e^{-i\pi/4} \\ e^{-i\pi/4} & -ie^{-i\pi/4} & ie^{-i\pi/4} & e^{-i\pi/4} \\ -ie^{i\pi/4} & e^{i\pi/4} & e^{i\pi/4} & ie^{i\pi/4} \end{pmatrix}. $$ I want to implement the CNOT gate, which is given by $$ \text{CNOT}=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}. $$ I am looking for an expression of CNOT in terms of $U_\text{entangle}$. In the most general form, this is given by $$ \text{CNOT} = A\: U_\text{entangle}\: B \:U_\text{entangle}\: C U_\text{entangle}\: D \dots U_\text{entangle} \: Z $$ where $A,B,C,\dots,Z$ are arbitrary combinations of 1-qubit gates. Hence even $A$ for instance can be composed of several 1-qubit gates.

I now want to find such matrices $A,\dots$ that give me a representation of $U_\text{entangle}$ in terms of CNOT and 1-qubit gates.

Is there an algorithm / a procedure to follow in order to determine the circuit that realizes CNOT? Is $U_\text{entangle}$ maybe a famous gate that has already been shown to produce CNOT with single-qubit unitaries? If the ''decomposing'' procedure doesn't follow a strict rule, are there at least any tips / rules of thumb you can give in order to find such a decomposition?

Any help is really appreciated! Many thanks in advance!

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You can use the KAK decomposition method described here.

And if you are using Qiskit, this method is implemented in TwoQubitBasisDecomposer class.

z = cmath.rect(1, np.pi / 4) # e^(iπ/4)

# Your two-qubit entangling gate as numpy array:
basis_unitary = 0.5 * np.array([
    [1j*z, z, z, -1j*z],
    [1/z, 1j/z, -1j/z, 1/z],
    [1/z, -1j/z, 1j/z, 1/z],
    [-1j*z, z, z, 1j*z]
])

decomposer = TwoQubitBasisDecomposer(UnitaryGate(basis_unitary))
circ = decomposer(CXGate().to_matrix())
circ.draw('mpl')

The resulting circuit:

enter image description here

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