Happy Almost Hallowe’en! Today we did two-qubit (and more-qubit) systems and met our first quantum gate, so here are a few problems in those directions.

Posters: don’t forget to work on your posters; check the Posters tab for the next deadline.

Relevant Textbook Sections:

Chapter 6 (Quantum): multiple qubits, entanglement, measuring one qubit of a multi-qubit state, and single-qubit gates.

To Do:

1. Consider the state $\frac{1}{\sqrt{2}}|01\rangle + \frac{1}{\sqrt{2}}|10\rangle$.
   1. What is the probability of measuring a $0$ in the first bit?
   2. What is the probability of measuring a $0$ in the second bit?
   3. What is the probability of measuring a $0$ in both bits? (I.e. after measuring both bits, what’s the probability you obtained $00$?)
2. For each of the following, determine if it is entangled or not (i.e. whether it can be written as a product of single-qubit states):
   1. $|111\rangle$
   2. $\frac{1}{\sqrt{2}}|10\rangle + \frac{1}{\sqrt{2}}|11\rangle$
   3. $\frac{1}{\sqrt{2}}|110\rangle + \frac{1}{\sqrt{2}}|111\rangle$
   4. $\frac{1}{\sqrt{2}}|010\rangle + \frac{1}{\sqrt{2}}|111\rangle$
3. In the state $\frac{1}{\sqrt{2}}|010\rangle + \frac{1}{\sqrt{2}}|111\rangle$, what is the probability of measuring a $0$ in the first qubit? middle qubit? last qubit?
4. Measure the following states in the first qubit and report the possible outcome bits, outcome states (the state of the system after the measurement), and probabilities. Don’t forget to normalize.
   1. the Bell state $\frac{1}{\sqrt{2}}|00\rangle + \frac{1}{\sqrt{2}}|11\rangle$
   2. $\frac{1}{\sqrt{14}}|01\rangle + \frac{2i}{\sqrt{14}}|10\rangle + \frac{3}{\sqrt{14}}|11\rangle$

Quantum gates. A bit of practice with the linear algebra behind gates:

5. Compute the complex conjugates of $3+7i$, $8$ and $8i$.
6. Compute the conjugate transpose of the complex matrix $\begin{pmatrix} 1 & 2+i \\ 3i & i \end{pmatrix}$. Check if this matrix is unitary.
7. Apply the Pauli $Z$ gate $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ to the single-qubit state $\frac{1+2i}{\sqrt{6}}|0\rangle + \frac{1}{\sqrt{6}}|1\rangle$. What is the output? Does measuring the output in the $|0\rangle$, $|1\rangle$ basis give different probabilities than measuring the input?
8. Consider the single-qubit states $|\psi\rangle = \frac{1}{\sqrt{2}}|0\rangle + \frac{1}{\sqrt{2}}|1\rangle$ and $|\phi\rangle = \frac{1}{\sqrt{2}}|0\rangle - \frac{1}{\sqrt{2}}|1\rangle$.
   1. Explain why, measuring the states in the basis $|0\rangle$, $|1\rangle$, you can’t tell the difference between these states.
   2. Now apply the Hadamard gate $H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$ to both states. 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?
9. Create a unitary single-qubit gate that takes $|0\rangle$ to a normalized version of $|0\rangle + i|1\rangle$.