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Compute the complex conjugates of $3+7i$, $8$ and $8i$.
Compute the conjugate transpose of the complex matrix $\begin{pmatrix} 1 & 2+i \\ 3i & i \end{pmatrix}$. Check if this matrix is unitary.
Apply the Pauli Z gate to the single-qubit state $\frac{1+2i}{\sqrt{6}}|0\rangle + \frac{1}{\sqrt{6}}|1\rangle$. What is the output?
Verify that the CNOT gate is unitary.
Apply the CNOT gate to the state $\frac{1+2i}{\sqrt{6}}|01\rangle + \frac{1}{\sqrt{6}}|10\rangle$.
Consider “applying the Hadamard gate the second qubit” as a two-qubit gate.
What does this do to the state $|00\rangle$?
What does this do to the state $|01\rangle$?
Write out the two-qubit 4×4 matrix that does this, and verify that it is unitary.
Begin with 2-qubit state $|00\rangle$.
Apply the CNOT gate the 2-qubit system and then the Hadamard gate the 2nd qubit.
Start over with $|00\rangle$. Now apply the Hadamard gate to the 2nd qubit and then the CNOT gate to the system.
Do you get the same answer?
Consider the single-qubit states $|\psi\rangle = \frac{1}{\sqrt{2}}|0\rangle + \frac{1}{\sqrt{2}}|1\rangle$ and $|\phi\rangle = \frac{1}{\sqrt{2}}|0\rangle – \frac{1}{\sqrt{2}}|1\rangle$.
Explain why, measuring the states in the basis $|0\rangle$, $|1\rangle$, you can’t tell the difference between these states.
Now apply the Hadamard gate $H$ to both states. Now explain what happens to $H|\psi\rangle$ and $H|\phi\rangle$ if you measure in the basis $|0\rangle$, $|1\rangle$. Can you tell the difference between these by measuring?
Create a unitary single-qubit gate that takes $|0\rangle$ to $|0\rangle + i|1\rangle$.