I am trying to learn some basics in quantum computing and reached a place where I need to understand deeply unitary matrix decomposition. Therefore, I am looking for some help whether literature or mathematical proofs to justify the reality of the assumption that I have.
I have seen somewhere, a H-gate and S-gate can be expressed in terms of rotated X, Y, Z.
The $T$ gate, being a single-qubit gate, can be decomposed into a product of rotations by Theorem 4.1 ($Z-Y$ decomposition for a single qubit) in Nielsen and Chuang. The $CNOT$, $SWAP$ and $CCNOT$ gates cannot be expressed as tensor products of single-qubit gates, which precludes expression as a product of single-qubit rotations.
If you could express them as products, then your qubits would never be entangled, and you could easily simulate your system classically by tracking the qubits independently.
Any single qubit gate can be decomposed to $X$, $Y$ and $Z$ rotations.
Alternatively, you can implement that gate with series of $T$ and $H$ gates using Solovay-Kitaev algorithm. However, this is tricky in practice and the algorithm is useful rather in theoretical parts of quantum computing.
It is also possible to use decomposition of single qubit gates to $Z$ rotations, $S$ and $H$ gates. See more in this thread.
Multi-qubit gates, like $CCNOT$, can be decomposed to single qubit gates and $CNOT$ gates. Note that $CNOT$ cannot be decomposed further. You can find more on gates decomposition, both single- and multi-qubit in this article.
