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
- "list the formulas in and explain with an example each"
- "you did not deliver proper examples but stayed abstract 1/v2 is not example but still a formula"
- "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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