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When I put each single qubit within a quantum register using Hadamard gates in superposition, how does it work that the whole register quantum-state is in superposition?
On the math-level the register-state is the tensor-product of the single qubit-states; but how is this realized on the physical level? How do the qubits interact? Looking on entanglement I take a CNOT and I know this gate is realized on the physical level. But what about the tensor-product on the physical level?
According to the Solovay-Kitaev theorm, the Clifford gates (which are generated by $H$, $S$ and $CNOT$), along with a non-Clifford gate - say the $T$ gate, can be used to approximate any unitary down to arbitrary precision. The state of the register is a pure-state so it can be “reached” using only these gates. Notice that the only 2 qubit gate here is the $CNOT$.
What this implies is that it is enough to physically implement the $CNOT$ gate as the only 2 qubit gate, (as well as the mentioned single qubit gates) to reach the entangled state of both registers.
I hope this andswers your question.
