|\psi\rangle = \alpha|0\rangle + \beta|1\rangle
Example: |\psi\rangle = \frac{1}{\sqrt{2}}|0\rangle + \frac{1}{\sqrt{2}}|1\rangle
This represents an equal superposition of |0⟩ and |1⟩ states.
|\Psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)
Example: |\Psi\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\0\\0\\1\end{pmatrix}
This is a two-qubit entangled state.
H = \frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1\\1 & -1\end{pmatrix}
Example: Applying H to |0⟩: H|0\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1\\1 & -1\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix} = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix}
U^\dagger U = UU^\dagger = I
Example: For the Pauli-X gate: X = \begin{pmatrix}0 & 1\\1 & 0\end{pmatrix} X^\dagger X = \begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix} = \begin{pmatrix}1 & 0\\0 & 1\end{pmatrix} = I
x = \begin{pmatrix}a\\b\end{pmatrix} = \begin{pmatrix}\frac{3}{5}\\\frac{4}{5}\end{pmatrix}
Example: Encoding as a quantum state: |\psi\rangle = a|0\rangle + b|1\rangle = \frac{3}{5}|0\rangle + \frac{4}{5}|1\rangle
⚠️ LLM-generated content notice: Parts of this page may have been created or edited with the assistance of a large language model (LLM). The prompts that have been used might be on the page itself, the discussion page or in straight forward cases the prompt was just "Write a mediawiki page on X" with X being the page name. While the content has been reviewed it might still not be accurate or error-free. at the end"