Analytic split of quantum gates as tensor networks in specific cases like Pauli rotations

[Qottmann]

I have a general mathematical / tensor network methods question:

I know from a previous discussion that a gate acting on two qubits can be applied to an MPS by turning it into an MPO with identities on the intermediate sites. E.g. take a \text{CNOT}_{i,j} acting on non-neighboring sites i and j. I can simply SVD the 4 \times 4 CNOT matrix to get two MPO matrices, i.e. the Q and R^\dagger from SVD:

Now I am wondering if there are some special cases where I can directly read-out the Q and R^\dagger matrix. In particular, I want to know this for Pauli rotations. Since we know that a Pauli rotation with some Pauli word \bigotimes_i P_i (think X_0 Y_3 Z_9)

U = \exp(-i \theta \bigotimes_i P_i) = \cos(\theta) \mathbb{I} -i \sin(\theta) \bigotimes_i P_i

has this specific form in terms of two sinusodials, I would think this is a low-entanglement inducing operation and there should be some efficient MPO decomposition.

I want to know if I can directly construct the MPO without having to slice the matrix using SVD. For gains but also because at a certain number of Pauli terms this becomes impossible (e.g. \exp(-i \theta X^{\otimes 50}).

Greatly appreciate any help! Thank you!

[mtfishman]

Maybe the easiest way would be to write the operator symbolically as an OpSum and then convert it to an MPO.

[Qottmann]

Oh yeah, didnt see the forest from all the trees Thanks @mtfishman !

For anyone finding this in the future, this is the MPO for Pauli rotations: