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My question is about the quantum computer. I have tried to prove that the quantum computer is equivalent to some logical circuit.
I know this has already been proven, but I will present my attempt:

  1. First we know that the quantum computer is composed of quantum logic gates https://en.wikipedia.org/wiki/Fredkin_gate. According to this link, this gate is universal

  2. So any quantum computer can be made up of those gates

  3. Now we make a logical circuit simulating the Fredkin gate

  4. Then each quantum circuit can be simulated by a logical circuit

Is this proof valid?

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It's not a valid proof (yet) because you haven't described how you plan to do step 3. You also haven't described how you plan to deal with superpositions of states.

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