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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:
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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
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The Identity matrix to not change a qubit.
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Pauli matrices. X is also called Bit-flip.
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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.:
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Teleportation
An Example quantum circuit:
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