Quantum computing

Notes

0 and 1 cbits (classical bits)

$$\left| 0 \right> = \begin{pmatrix} 1\\ 0 \end{pmatrix}$$

$$\left| 1 \right> = \begin{pmatrix} 0\\ 1 \end{pmatrix}$$

• Quantum computers only use reversible operations
• Identity and Negation are reversible
• Constant-0 and Constant-1 aare not reversible

Tensor product of vectors

$$\begin{pmatrix} x_0\\x_1 \end{pmatrix} \otimes \begin{pmatrix} y_0\\y_1 \end{pmatrix} = \begin{pmatrix} x_0 \begin{pmatrix} y_0\\y_1 \end{pmatrix}\\x_1 \begin{pmatrix} y_0\\y_1 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} x_0y_0\\ x_0y_1\\ x_1y_0\\ x_1y_1 \end{pmatrix}$$

$$\begin{pmatrix} 1\\2 \end{pmatrix} \otimes \begin{pmatrix} 3\\4 \end{pmatrix} = \begin{pmatrix} 3\\ 4\\ 6\\ 8 \end{pmatrix}$$

Multiple cbits representation

• This tensored representation is called the product state

$$\left| 00 \right> = \begin{pmatrix} 1\\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1\\ 0 \end{pmatrix} = \begin{pmatrix} 1\\ 0\\ 0\\ 0\\ \end{pmatrix}$$

$$\left| 01 \right> = \begin{pmatrix} 1\\ 0 \end{pmatrix} \otimes \begin{pmatrix} 0\\ 1 \end{pmatrix} = \begin{pmatrix} 0\\ 1\\ 0\\ 0\\ \end{pmatrix}$$

$$\left| 10 \right> = \begin{pmatrix} 0\\ 1 \end{pmatrix} \otimes \begin{pmatrix} 1\\ 0 \end{pmatrix} = \begin{pmatrix} 0\\ 0\\ 1\\ 0\\ \end{pmatrix}$$

$$\left| 11 \right> = \begin{pmatrix} 0\\ 1 \end{pmatrix} \otimes \begin{pmatrix} 0\\ 1 \end{pmatrix} = \begin{pmatrix} 0\\ 0\\ 0\\ 1\\ \end{pmatrix}$$

$$\left| 4 \right> = \left| 100 \right> = \begin{pmatrix} 0\\ 1 \end{pmatrix} \otimes \begin{pmatrix} 1\\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1\\ 0 \end{pmatrix} = \begin{pmatrix} 0\\ 0\\ 0\\ 0\\ 1\\ 0\\ 0\\ 0\\ \end{pmatrix}$$

CNOT

$$C = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ \end{pmatrix}$$

$$C\left| 10 \right> = C \begin{pmatrix} \begin{pmatrix} 0\\ 1\\ \end{pmatrix} \otimes \begin{pmatrix} 1\\ 0\\ \end{pmatrix} \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ \end{pmatrix} \begin{pmatrix} 0\\ 0\\ 1\\ 0\\ \end{pmatrix} = \begin{pmatrix} 0\\ 0\\ 0\\ 1\\ \end{pmatrix} = \begin{pmatrix} 0\\ 1\\ \end{pmatrix} \otimes \begin{pmatrix} 0\\ 1\\ \end{pmatrix} = \left| 11 \right>$$

References

• https://www.youtube.com/watch?v=F_Riqjdh3oM