vault backup: 2026-07-30 09:29:48
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Benjamin Neumann
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"file": "SS2026/Quantum Computing/Introduction to Quantum Computing.md",
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"title": "Introduction to Quantum Computing"
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"title": "Quantum Circuits"
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@@ -0,0 +1,48 @@
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Now we look at circuits constructed with what we looked at.
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These systems will consist of qubits $\in \mathbb{C}^2$ .
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A very simple example for a quantum circuit, now using pictures instead of formulas:
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![[Pasted image 20260730091827.png]]
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Here we have two wires 0 and +.
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First we apply the unitaries X and H to the wires, written as $X \otimes H$.
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Then we apply the unitary U to both qubits (wires) .
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Finally we apply H to the second wire again, because we don't do anything
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with the top wire, we apply the identity to it so $I \otimes H$
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Last but not least we measure the top wire.
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## Important gates
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We distinguish between gates on single qubits and gates on multiple qubits.
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We will introduce some of the more important ones:
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### Single qubit gates
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![[Pasted image 20260730092306.png]]
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The Identity matrix to not change a qubit.
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![[Pasted image 20260730092336.png]]
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Pauli matrices. X is also called Bit-flip.
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![[Pasted image 20260730092409.png]]
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The Hadamard-gate transforms a classical bit $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and transforms it into a
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superposition $\begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{pmatrix}$
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### The CNOT Gate
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The controlled-NOT gate operates on two qubits.:
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![[Pasted image 20260730092639.png]]
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![[Pasted image 20260730092648.png]]
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## Teleportation
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An Example quantum circuit:
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![[Pasted image 20260730092802.png]]
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![[Pasted image 20260730092809.png]]
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![[Pasted image 20260730092817.png]]
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