Update QuBit.md

docs/QuBit/QuBit.md

@@ -1,2 +1,197 @@
 ## QuBit
 
+QuBit(alpha: complexe, beta: complexe)
+
+- alpha: complexe value for |0>
+- beta: complexe value for |1>
+
+### Attributes
+
+- ```__m: list[list[complexe]]```
+
+  the values contained in the matrix.
+
+- ```__size: tuple[int, int]```
+
+  the size of the matrix
+
+### Methode
+
+- ```__init__(l: list[list[complex]]=[]) -> None```
+
+  initiates the matrix
+
+- ```__mul__(__value: "Matrix") -> Matrix```
+
+  defines the natural multiplication as the Koeneger product for matrices
+
+- ```__apply(x: list[complex]) -> list[complex]```
+
+  takes a list to represent a vector in a colonne, and return the proctuct of the matrix by this vector
+
+- ```__str__() -> str```
+
+  returns a string representation of the matrix
+
+
+### Staticmethods
+
+- ```I() -> Matrix```
+
+  returns the matrix
+
+```math
+\begin{pmatrix}
+1 & 0\\
+0 & 1
+\end{pmatrix}
+```
+
+- ```H() -> Matrix```
+
+  returns the matrix
+
+```math
+\begin{pmatrix}
+\frac {1} {\sqrt 2} & \frac {1} {\sqrt 2}\\
+\frac {1} {\sqrt 2} & -\frac{1} {\sqrt 2}
+\end{pmatrix}
+```
+
+- ```X() -> Matrix```
+
+  returns the matrix
+
+```math
+\begin{pmatrix}
+0 & 1\\
+1 & 0
+\end{pmatrix}
+```
+
+- ```Y() -> Matrix```
+
+  returns the matrix
+
+```math
+\begin{pmatrix}
+0 & -i\\
+i & 0
+\end{pmatrix}
+```
+
+- ```Z() -> Matrix```
+
+  returns the matrix
+
+```math
+\begin{pmatrix}
+1 & 0\\
+0 & -1
+\end{pmatrix}
+```
+
+- ```S() -> Matrix```
+
+  returns the matrix
+
+```math
+\begin{pmatrix}
+1 & 0\\
+0 & i
+\end{pmatrix}
+```
+
+- ```T() -> Matrix```
+
+  returns the matrix
+
+```math
+\begin{pmatrix}
+1 & 0\\
+0 & e^{i\frac {\pi} {4}}
+\end{pmatrix}
+```
+
+- ```Rx(phi: float) -> Matrix```
+
+  returns the matrix
+
+```math
+\begin{pmatrix}
+\cos(\frac \phi 2) & -i\sin(\frac \phi 2)\\
+-i\sin(\frac \phi 2) & \cos(\frac \phi 2)
+\end{pmatrix}
+```
+
+- ```Ry(phi: float) -> Matrix```
+
+  returns the matrix
+
+```math
+\begin{pmatrix}
+\cos(\frac \phi 2) & -\sin(\frac \phi 2)\\
+\sin(\frac \phi 2) & \cos(\frac \phi 2)
+\end{pmatrix}
+```
+
+- ```Rz(phi: float) -> Matrix```
+
+  returns the matrix
+
+```math
+\begin{pmatrix}
+e^{-i\frac \phi 2} & 0\\
+0 & e^{i\frac \phi 2}
+\end{pmatrix}
+```
+
+- ```R1(phi: float) -> Matrix```
+
+  returns the matrix
+
+```math
+\begin{pmatrix}
+1 & p\\
+0 & e^{i\phi}
+\end{pmatrix}
+```
+
+- ```CNOT() -> Matrix```
+
+  returns the matrix
+
+```math
+\begin{pmatrix}
+1 & 0 & 0 & 0\\
+0 & 1 & 0 & 0\\
+0 & 0 & 0 & 1\\
+0 & 0 & 1 & 0
+\end{pmatrix}
+```
+
+- ```SWAP() -> Matrix```
+
+  returns the matrix
+
+```math
+\begin{pmatrix}
+1 & 0 & 0 & 0\\
+0 & 0 & 1 & 0\\
+0 & 1 & 0 & 0\\
+0 & 0 & 0 & 0
+\end{pmatrix}
+```
+
+- ```Cu(u: Matrix) -> Matrix```
+
+  returns the matrix
+
+```math
+\begin{pmatrix}
+1 & 0 & 0 & 0\\
+0 & 1 & 0 & 0\\
+0 & 0 & u_{00} & u_{01}\\
+0 & 0 & u_{10} & u_{11}
+\end{pmatrix}
+```