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My question is very basic but I was still unable to find its correct answer. I am studying quantum gates from various books. They calculate eigenvalues and eigenvectors with quantum gates.

What is the relevance of calculating eigenvalue and eigenvector? Also, I read a lot that by putting Hadamard gate, the qubit is put into superposition. As per my understanding, the qubit comes into the equatorial region of the Bloch sphere in superposition. What exactly does it mean and what is its benefit?

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    $\begingroup$ each post should contain a single, laser-focus question. You can open different posts to ask different questions. Feel free to edit this post accordingly. You can also have a look at quantumcomputing.stackexchange.com/help/how-to-ask $\endgroup$
    – glS
    Apr 7, 2022 at 15:23
  • $\begingroup$ Thank you @gIS for editing my question. $\endgroup$
    – Manu
    Apr 8, 2022 at 3:38

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Check here for eigenvectors and eignvalues.

As for superposition and entanglement, they are both quantum phenomena. When a qubit is in superposition it means that it described by a linear combination of the basis states $|0\rangle$ and $|1\rangle$. So in your example, after applying a Hadamard gate on qubit in basis state $|0\rangle$, the qubit will now have an equal probability of being 0 or 1. After measuring the state will collapse to only 0 or 1. Entanglement is another phenomenon. In simple terms it means that 2 or more qubit values become correlated even if they were separated by large distances.

Both phenomena are used heavily in quantum algorithms and are the main differences between classical and quantum computing.

You can also check here for more details on Superposition and Entanglement

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  • $\begingroup$ Thank you @kmagdy $\endgroup$
    – Manu
    Apr 8, 2022 at 3:49
  • $\begingroup$ Why entanglement is used heavily in quantum computing and what is the exact benefit we get from entanglement? $\endgroup$
    – Manu
    Apr 8, 2022 at 3:51
  • $\begingroup$ There are a lot of algorithms that rely on it, The simplest example is quantum teleportation {algorithm](qiskit.org/textbook/ch-algorithms/teleportation.html) $\endgroup$
    – kibrahim
    Apr 8, 2022 at 8:57

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