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Exponential representation of single-qubit gates

For arbitrary single-qubit gate $U_{\mathbf{1}}$, we can express it in such a form

$$ U_{\mathbf{1}} = e^{i\alpha}\exp[-i\frac{\theta}{2}\vec{v}\cdot\vec{\sigma}]= e^{i\alpha}\exp[-i\frac{\theta}{2}(xX+yY+zZ)], $$

where $\alpha$ is the global phase, $\theta$ is the rotation angle of $U$ on the Bloch sphere, $\vec{v}=(x,y,z)$ is a real 3-d vector with length equal to 1, and $\vec{\sigma}=(X, Y, Z)$ stand for Pauli matrix vector. This fact can be found in Exercise 4.8 from the Nielson and Chuang's textbook.

Generalization to two-qubit gates

Similarly, I am wondering if we have a conclusion for two-qubit gates like

$$ U_{\mathbf{2}} = e^{i\alpha}\exp[-i\frac{\theta}{2}\vec{v_{\mathbf{2}}}\cdot\vec{\sigma_{\mathbf{2}}}], $$

where I guess $\vec{v_{\mathbf{2}}}$ is a real 15-d vector (as we have 15 non-trivial two-qubit Pauli bases), and $\vec{\sigma_{\mathbf{2}}} = (IX, IY, IZ, XI, XX..., ZZ)$ is the Pauli matrix vector with 15 elements. Does there exist such a generalization? If the answer is no, can we still define rotation angle for arbitrary two-qubit gates?

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I think what you want is the KAK decomposition. An available implementation is cirq.kak_decomposition.

The KAK decomposition decomposes an arbitrary two-qubit unitary gate $U_2$ into single qubit gates and same-basis Pauli coupling terms:

$$U_2 = (A \otimes B) \cdot (X \otimes X)^x \cdot (Y \otimes Y)^y \cdot (Z \otimes Z)^z \cdot (C \otimes D) e^{i\theta}$$

where $A, B, C, D \in \mathbb{SU}(2)$ and $x, y, z, \theta \in \mathbb{R}$.

For two-qubit Clifford gates the single qubit gates will be Cliffords and the coupling terms can be simplified to the presence or absence of a CZ and a SWAP. So two-qubit Cliffords gates can be broken down into ones that are CZ-like, SWAP-like, CZSWAP-like, and identity-like.

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