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2

It cannot be the case that $H=e^{i\pi/4}\sqrt{iNOT}$. Whatever your interpretation of $iNOT$ (I'd agree with your definition), just square the thing. $H^2=I$, the identity, and so it is certainly not the case that $$I=e^{i\pi/2}iNOT.$$ It is true, however, that if you perform the sequence that would give you an operation such as $NOT$ or $iNOT$, and you ...

4

The coefficient $\frac{1}{\sqrt{2^3}}$ is the normalization factor: if you have a 3-qubit state that is an equal superposition of 8 basis states, its norm still has to be 1; thus the squared amplitude of each basis state has to be $\frac{1}{8}$, and the amplitude will be square root of that, i.e. exactly $\frac{1}{\sqrt{2^3}}$. You can also obtain that ...

4

Hadamard gate can be interpreted as a rotation in 3D Euclidean space (on Bloch sphere) by angle $\pi$ around X+Z axis. The qubit rotation by angle $\theta$ around axis pointed by unit vector $\textbf{n}=\{n_x,n_y,n_z\}$ is described by rotation operator ($X$, $Y$ and $Z$ are Pauli matrices) \begin{align} R_{\textbf{n}}(\theta)=&n_xe^{-i\frac{\theta}{2}X}...

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The first postulate of quantum mechanics that can be found in the M. Nielsen and I. Chuang textbook: Postulate 1: Associated to any isolated physical system is a complex vector space with the inner product (that is, a Hilbert space) known as the state space of the system. The system is completely described by its state vector, which is a unit vector in the ...

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