Now we look at circuits constructed with what we looked at.
These systems will consist of qubits $\in \mathbb{C}^2$ .

A very simple example for a quantum circuit, now using pictures instead of formulas:

![[Pasted image 20260730091827.png]]

Here we have two wires 0 and +.
First we apply the unitaries X and H to the wires, written as $X \otimes H$.
Then we apply the unitary U to both qubits (wires) .
Finally we apply H to the second wire again, because we don't do anything 
with the top wire, we apply the identity to it so $I \otimes H$ 
Last but not least we measure the top wire.


## Important gates

We distinguish between gates on single qubits and gates on multiple qubits.
We will introduce some of the more important ones:

### Single qubit gates

![[Pasted image 20260730092306.png]]
The Identity matrix to not change a qubit.

![[Pasted image 20260730092336.png]]
Pauli matrices. X is also called Bit-flip.

![[Pasted image 20260730092409.png]]
The Hadamard-gate transforms a classical bit $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and transforms it into a
superposition $\begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{pmatrix}$ 


### The CNOT Gate
The controlled-NOT gate operates on two qubits.:

![[Pasted image 20260730092639.png]]
![[Pasted image 20260730092648.png]]



## Teleportation
An Example quantum circuit:

![[Pasted image 20260730092802.png]]
![[Pasted image 20260730092809.png]]
![[Pasted image 20260730092817.png]]