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