qcheatsheet
Basic operation
- \(\left| \Psi \right> = \alpha \left| 0 \right> + \beta \left| 1 \right>\)
- \(\left| \Psi \right> = cos(\frac{\theta}{2}) \left| 0 \right> + e^{i\phi} sin(\frac{\theta}{2}) \left| 1 \right>\)
- \(\left| + \right> = \frac{1}{\sqrt{2}} (\left| 0 \right> + \left| 1 \right>)\)
- \(\left| - \right> = \frac{1}{\sqrt{2}} (\left| 0 \right> - \left| 1 \right>)\)
QGates
PauliX - NOT - \(\sigma_x\)
- Apply a 180º rotation around X axis
- swap α and β probabilities and keep the same phase ϕ.
- Effect on base:
$$
\left< \sigma_x | 0 \right> = \left| 1 \right>
\\
\left< \sigma_x | 1 \right> = \left| 0 \right>
$$
- Matrix:
$$
\begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}
$$
PauliY - \(\sigma_y\)
- Apply a 180º rotation around Y axis
- Swap α and β probabilities and shift the phase ϕ.
- Effect on base
$$
\begin{align}
&\left< \sigma_y | 0 \right> = i\left| 1 \right>
\\
&\left< \sigma_y | 1 \right> = -i\left| 0 \right>
\end{align}
$$
- Matrix
$$
\begin{bmatrix}
0 & -i \\
i & 0
\end{bmatrix}
$$
PauliZ - \(\sigma_z\)
- Apply a 180º rotation around Z axis
- Keep α and β probabilities and flip the phase ϕ.
- Effect on base:
$$
\begin{align}
&\left< \sigma_z | 0 \right> = \left| 0 \right>
\\
&\left< \sigma_z | 1 \right> = -\left| 1 \right>
\end{align}
$$
- Matrix
$$
\begin{bmatrix}
1 & 0 \\
0 & -1
\end{bmatrix}
$$
Hadamard - \(H\)
- Apply a 90º rotation around the Y-axis, followed by a 180º rotation around the X-axis
- Effect on base:
$$
\begin{align}
&\left< H | 0 \right> = \frac{1}{\sqrt{2}} ( \left| 0 \right> + \left| 1 \right>) = \left| + \right>\\
&\left< H | 1 \right> = \frac{1}{\sqrt{2}} ( \left| 0 \right> - \left| 1 \right>) = \left| - \right>\\
\end{align}
$$
- Matrix:
$$
\frac{1}{\sqrt{2}}
\begin{bmatrix}
1 & 1 \\
1 & -1
\end{bmatrix}
$$
Tensor product - \(\otimes\)
- \(\left| a b \right> = \left| a \right> \otimes \left| b \right>\)
- \(\lambda \left| ab \right> = \lambda \left| a \right> \otimes \left| b \right> = \left| a \right> \otimes \lambda \left| b \right>\)
- \((\left| a \right> + \left| b \right>) \otimes \left| c \right> = \left| a \right> \otimes \left| c \right> + \left| b \right> \otimes \left| c \right> = \left| a c \right> + \left| b c \right>\)