Formula Explanation

Formulas from the paper

Superposition of quantum states

[math]|\psi\rangle = \alpha|0\rangle + \beta|1\rangle[/math]

Example: [math]|\psi\rangle = \frac{1}{\sqrt{2}}|0\rangle + \frac{1}{\sqrt{2}}|1\rangle[/math]

This represents an equal superposition of |0⟩ and |1⟩ states.

Bell's state

[math]|\Psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)[/math]

Example: [math]|\Psi\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\0\\0\\1\end{pmatrix}[/math]

This is a two-qubit entangled state.

Hadamard gate

[math]H = \frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1\\1 & -1\end{pmatrix}[/math]

Example: Applying H to |0⟩: [math]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}[/math]

Unitary operator condition

[math]U^\dagger U = UU^\dagger = I[/math]

Example: For the Pauli-X gate: [math]X = \begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}[/math] [math]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[/math]

Amplitude encoding

[math]x = \begin{pmatrix}a\\b\end{pmatrix} = \begin{pmatrix}\frac{3}{5}\\\frac{4}{5}\end{pmatrix}[/math]

Example: Encoding as a quantum state: [math]|\psi\rangle = a|0\rangle + b|1\rangle = \frac{3}{5}|0\rangle + \frac{4}{5}|1\rangle[/math]

Prompts used to obtain this result

1. "list the formulas in and explain with an example each"
2. "you did not deliver proper examples but stayed abstract 1/v2 is not example but still a formula"
3. "now i still do not have the result i asked for a list of formulas of the paper + examples. Give me the result in mediawiki format using the simplemathjax extensions notation and start with listing the prompts i had to give to get the results add an

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