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In the quantum mechanics we are taught first at uni, we learn about schrödinger equation, $i \hbar \partial_t |\Psi\rangle = \hat H |\Psi\rangle,$ eigenvectors $|\Psi \rangle$, and operators on the like the momentum operator, $\hat p = -i\hbar\partial_x$ and the time evolution operator $\hat U = \exp[-it/\hbar \hat H]$, which is unitary.

When i read about quantum computing, the eigenvectors are there, just the same, however the operators being talked about are the "gates", like the hadamard gate, which are unitary matrices. How are they related to the "traditional" quantum mechanics? Are they different time-evolution operators for a quantum mechanical system?

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    $\begingroup$ I would recommend you to watch this if you have time youtube.com/… Gives quite a nice explanation on the unitarity of operators and their relation to quantum gates. $\endgroup$ – Tachyon209 Apr 9 at 12:08
  • $\begingroup$ Thank you, @Tachyon209, that is a great video! $\endgroup$ – Martin Johnsrud Apr 9 at 18:11
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The gates are unitary transformations that allow one to implement logical transformations to your qubits. Since they are unitary, they correspond to time evolution under some Hamiltonian.

You could simply take the $\log$ of the gate and find a valid Hamiltonian, in actual implementations the physical system used is engineered in such a way that you can tune the Hamiltonian to create different time evolutions corresponding to the different gates.

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