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3

The black (white) dot means a condition that the corresponding qubit should be in $|1\rangle$ state ($|0\rangle$ state) in order to apply the gate. The first circuit implements the Hadamard gate only if the first qubit is in $|1\rangle$ state and the second qubit is in $|0 \rangle$ state (similar discussions can be found here). In other words, if the ...

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I'll try to explain the whole process here, I hope this will clear the equation for you! So, what we are trying to do here is to find a matrix that represents the linear operator $A : V \rightarrow W$. $V$ and $W$ are both vector spaces so we know we can find one basis for each set, here $|v_1\rangle, ..., |v_m\rangle$ and $|w_1\rangle, ..., |w_n\rangle$. ...

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For measuring 2 qubits, there are 4 possible outcomes, corresponding to projectors $$P_{00}=|00\rangle\langle 00|,\qquad P_{01}=|01\rangle\langle 01|,\qquad P_{10}=|10\rangle\langle 10|,\qquad P_{11}=|11\rangle\langle 11|.$$ So, if you have a state $|\psi\rangle$, you get the outcome $x$ with probability $p_x=\langle\psi|P_x|\psi\rangle$. This immediately ...

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My understanding of the OP's question is that there is some restriction imposed that a gates can only act on Adjacent Qubits. While this isn't necessary, we can still work with this restriction using SWAP gates to make non-adjacent qubit adjacent. If the Control qubits are $i$ and $j$; and target qubit is $k$. Such that $i+1<j$ and $j+1<k$. Then we can ...

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