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+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]]
+