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I am currently going through Nielsen's QC bible and having still some foundational / conceptual problems with the matter.

Nielsen Quantum Computing

I have tried to retrieve this $8 {\times} 8$ matrix describing the QFT of 3 qubits via Kronecker product in various attempts.

Hadamard transform can be decomposed into $H \otimes 1 \otimes 1$, and the others are fundamentally kronecker products of the 4x4 matrices of S resp. T with the 2x2 identity.

Whats wrong with my approach?

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Your approach is fine. One thing that might be tripping you up is to make sure that the $T$ is operating on the first and third tensor factors. It is easier when the tensor factors are right next to each other so you can write $T$ operating on $2$ and $3$ as KroneckerProduct[IdentityMatrix[2],T] in Mathematica. So to handle $1$ and $3$ you would do

KroneckerProduct[Swap,IdentityMatrix[2]]
  .KroneckerProduct[IdentityMatrix[2],T]
  .KroneckerProduct[Swap,IdentityMatrix[2]]

Here you are giving the explicit $4{\times}4$ matrices for Swap and T. Similarly make sure to do KroneckerProduct[IdentityMatrix[2],H,IdentityMatrix[2]] for the second H.

Keep track of indices when doing the Kronecker products and don't do a matrix bigger than $3 {\times} 3$ by hand.

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