A gate for a Quantum Computer

It is like a beam splitter. Let HH be an Operator where

H=12(111−1)=Z eiπ4YH = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} = Z\,e^{i\frac{\pi}{4}Y}

It creates Superposition where ∣±⟩\ket{\pm} is defined in Spin-1/2 System

H∣0⟩=∣+⟩,H∣1⟩=∣−⟩H\ket{0} = \ket{+}, \qquad H\ket{1} = \ket{-}

Fact about on the basis change identities:

HZH=X,HXH=ZHZH = X, \qquad HXH = Z