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I was watching the following video https://www.youtube.com/watch?v=IrbJYsep45E
And around 3 minutes in, they do an example computation with two qubits. The qubits started off in the state $|00\rangle$, and then go through what I assume is a Hadamard gate which puts them in a superposition. Why is it a $50\%$ chance of being $|01\rangle$, and a fifty percent chance of being $|10\rangle$? Why is it not a quarter chance of being $|00\rangle$, $|10\rangle$, $|01\rangle$, and $|11\rangle$?

Sanchayan Dutta
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John Jepson
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  • It's fair to wonder why they describe quantum gates that way, but they don't actually suggest that it's a Hadamard gate that they're applying, or even any gate in particular; they just say "a quantum gate". They then follow it up with another seemingly arbitrary and anonymous "quantum gate". In fairness, that's better than many treatments I've seen in that they're using real math with actual irrational numbers, but they aren't doing an especially good job at explaining themselves at that stage. – Niel de Beaudrap Oct 19 '18 at 23:34

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There is a slight misconception in your question. You can't apply a Hadamard gate on $|00\rangle$. Because the basis state $|00\rangle$ is actually a tensor product of a $2\times 1$ dimensional basis state $|0\rangle$ used twice. This results in $|00\rangle = |0\rangle \otimes |0\rangle = \begin{bmatrix}1 \\0\\0\\0\end{bmatrix}$. Now you must take the tensor product of Hadamard gate(which is $2\times 2$ dimensional) too in order to apply it in this state. Like so: $$H \otimes H (|0\rangle \otimes |0\rangle)$$ $$= H|0\rangle \otimes H|0\rangle$$ $$ =\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) \otimes \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$$ $$=\frac{1}{2}(|00\rangle + |01\rangle + |10\rangle + |11\rangle)$$. Now obviously $ 4 \times (\frac{1}{2})^2 = 1$, as you have correctly inferred.

Sanchayan Dutta
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QuestionEverything
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It is not two Hadamard gates.
You are 100% correct that two Hadamrd gates would put you in an equal superposition of 00, 01, 10, 11, meaning 25% chance of getting any of those.

Let's construct her gate based on the input $|00\rangle$ and the output $\frac{1}{\sqrt{2}}\left(|10\rangle +|01\rangle\right)$.

The final state is $|\Psi^+\rangle$ from this question.

So the state can be constructed with this circuit:

enter image description here

where the middle gate is a CNOT.

user1271772 No more free time
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