Quantum Computing

incomplete Conventional Computing has binary digits (0,1)

Transistor:

Either something goes through or not.

You use the fact that quantum mechanical systems are a linear combination of many states. Instead of just 2 bits, you can have a large number of “qubits” that arise from the linear superposition of the two level states.

• 2 level systems denoted by ${\chi }_{+}=\left(\begin{array}{c}1\\ 0\end{array}\right)$ and ${\chi }_{-}=\left(\begin{array}{c}0\\ 1\end{array}\right)$

Recall:

$${\chi }_{+}={\chi }_{sm}$$ $$\psi =\alpha {\chi }_{+}+\beta {\chi }_{-}$$

Block Sphere describes a superposition of states.

$$\psi =\cos \frac{\theta }{2}+e^{i\varphi }\sin \frac{\theta }{2}$$

in matrix notation:

$$\psi =\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 0\end{array}\right)+\frac{1}{\sqrt{2}}\left(\begin{array}{c}0\\ 1\end{array}\right)$$

In class 2 & 3

$$\theta =e^{i\varphi };\varphi =0$$