>>16725416 (OP)
test
[eqn]
|\psi_0\rangle = |0\rangle \otimes |000\rangle
|\psi_1\rangle = H|0\rangle \otimes |000\rangle = \frac{1}{\sqrt{2}} \left( |0\rangle \otimes |000\rangle + |1\rangle \otimes |000\rangle \right)
|\psi_2\rangle = \frac{1}{\sqrt{2}} \left( |0\rangle \otimes |000\rangle + |1\rangle \otimes |101\rangle \right)
H^{\otimes 3} |000\rangle = \frac{1}{\sqrt{8}} \sum_{x \in {0,1}^3} |x\rangle
H^{\otimes 3} |101\rangle = \frac{1}{\sqrt{8}} \sum_{x \in {0,1}^3} (-1)^{x_0 + x_2} |x\rangle
|\psi_{\text{final}}\rangle = \frac{1}{\sqrt{2}} \left(
|0\rangle \otimes \frac{1}{\sqrt{8}} \sum_{x \in {0,1}^3} |x\rangle +
|1\rangle \otimes \frac{1}{\sqrt{8}} \sum_{x \in {0,1}^3} (-1)^{x_0 + x_2} |x\rangle
\right)
= \sum_{x \in {0,1}^3} \frac{1}{\sqrt{16}} \left(
|0\rangle \otimes |x\rangle + (-1)^{x_0 + x_2} |1\rangle \otimes |x\rangle
\right)
[/eqn]
