Quantum Computation and Quantum Error Prevention

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Here's a random test of math functions (from quantiki.org):

Let's see, let <left>${\left\vert {\psi }\right\rangle }=\alpha {\left\vert {0}\right\rangle }+\beta {\left\vert {1}\right\rangle }$</left> then

${\left\vert {\psi }\right\rangle }\otimes \beta _{00}$ $=(\alpha {\left\vert {0}\right\rangle }+\beta {\left\vert {1}\right\rangle })\left({\frac {1}{\sqrt {2}}}({\left\vert {00}\right\rangle }+{\left\vert {11}\right\rangle })\right)$ $={\frac {1}{\sqrt {2}}}\left(\alpha {\left\vert {0}\right\rangle }({\left\vert {00}\right\rangle }+{\left\vert {11}\right\rangle })+\beta {\left\vert {1}\right\rangle }({\left\vert {00}\right\rangle }+{\left\vert {11}\right\rangle })\right)$ ${\;{{CNOT(1,2)} \atop \longrightarrow }\;}{\frac {1}{\sqrt {2}}}\left(\alpha {\left\vert {0}\right\rangle }({\left\vert {00}\right\rangle }+{\left\vert {11}\right\rangle })+\beta {\left\vert {1}\right\rangle }({\left\vert {10}\right\rangle }+{\left\vert {01}\right\rangle })\right)$ ${\;{{H(1)} \atop \longrightarrow }\;}{\frac {1}{\sqrt {2}}}\left(\alpha {\frac {1}{\sqrt {2}}}({\left\vert {0}\right\rangle }+{\left\vert {1}\right\rangle })({\left\vert {00}\right\rangle }+{\left\vert {11}\right\rangle })+\beta {\frac {1}{\sqrt {2}}}({\left\vert {0}\right\rangle }-{\left\vert {1}\right\rangle })({\left\vert {10}\right\rangle }+{\left\vert {01}\right\rangle })\right)$ $={\frac {1}{2}}\left({\left\vert {00}\right\rangle }(\alpha {\left\vert {0}\right\rangle }+\beta {\left\vert {1}\right\rangle })+{\left\vert {01}\right\rangle }(\alpha {\left\vert {1}\right\rangle }+\beta {\left\vert {0}\right\rangle })+{\left\vert {10}\right\rangle }(\alpha {\left\vert {0}\right\rangle }-\beta {\left\vert {1}\right\rangle })+{\left\vert {11}\right\rangle }(\alpha {\left\vert {1}\right\rangle }-\beta {\left\vert {0}\right\rangle })\right)$ $={\frac {1}{2}}\sum _{b_{1}b_{2}=0}^{1}{\left\vert {b_{1}b_{2}}\right\rangle }(X^{b_{2}}Z^{b_{1}}){\left\vert {\psi }\right\rangle }$