1. If $x=\frac{1}{2}$, which of the following vectors can be a valid quantum state?

    a) $\left(\begin{array}{c}x\end{array}\right)$    b) $\left(\begin{array}{c}x\\ x\end{array}\right)$    c) $\left(\begin{array}{c}x\\ x\\ x\end{array}\right)$    d) $\left(\begin{array}{c}x\\ x\\ x\\ x\end{array}\right)$    e) $\left(\begin{array}{c}x\\ x\\ x\\ x\\ x\end{array}\right)$        

2. If $|u⟩=\left(\begin{array}{c}\frac{1}{2}\\ x\\ y\end{array}\right)\in {\mathbb{R}}^3$ is a quantum state, which one of the following equations cannot be possible?

    a) $x+y=\frac{1}{2}$    b) $x+y=\frac{3}{4}$    c) $x-y=0$    d) $x^2+y^2=\frac{1}{2}$    e) $x^2+y^2=\frac{3}{4}$        

3. We have a three state quantum system. If the system is in the quantum state $|u⟩=\left(\begin{array}{c}\frac{1}{3}-\frac{1}{\sqrt{3}}\\ \frac{1}{3}+\frac{1}{\sqrt{3}}\\ x\end{array}\right)\in {\mathbb{R}}^3$, what is the probability of being in the third state?

    a) $\frac{1}{9}$    b) $\frac{1}{8}$    c) $\frac{1}{6}$    d) $\frac{1}{3}$    e) $\frac{1}{2}$        

4. If $|u⟩\in {\mathbb{R}}^2$ is a quantum state on the unit circle with angle is $\frac{2\pi }{3}$, what is $|u⟩$?

    a) $\left(\begin{array}{r}\frac{1}{2}\\ \frac{\sqrt{3}}{2}\end{array}\right)$    b) $\left(\begin{array}{r}-\frac{1}{2}\\ \frac{\sqrt{3}}{2}\end{array}\right)$    c) $\left(\begin{array}{r}\frac{\sqrt{3}}{2}\\ \frac{1}{2}\end{array}\right)$    d) $\left(\begin{array}{r}-\frac{\sqrt{3}}{2}\\ \frac{1}{2}\end{array}\right)$    e) $\left(\begin{array}{r}-\frac{\sqrt{3}}{2}\\ -\frac{1}{2}\end{array}\right)$        

5. What is $H^4$, where $H$ is Hadamard operator?

    a) $\left(\begin{array}{rr}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\end{array}\right)$    b) $\left(\begin{array}{rr}\frac{1}{2}& \frac{1}{2}\\ \frac{1}{2}& -\frac{1}{2}\end{array}\right)$    c) $\left(\begin{array}{rr}1& 0\\ 0& 1\end{array}\right)$    d) $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$    e) $\left(\begin{array}{rr}1& 0\\ 0& -1\end{array}\right)$        

6. What is $H^7$?

    a) $\left(\begin{array}{rr}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\end{array}\right)$    b) $\left(\begin{array}{rr}\frac{1}{2}& \frac{1}{2}\\ \frac{1}{2}& -\frac{1}{2}\end{array}\right)$    c) $\left(\begin{array}{rr}1& 0\\ 0& 1\end{array}\right)$    d) $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$    e) $\left(\begin{array}{rr}1& 0\\ 0& -1\end{array}\right)$        

7. We have a qubit in state $|0⟩$. We apply the operators $H,X,X,H,X$ in order, where $X$ is NOT operator. What is the final state?

    a) $|0⟩$    b) $|1⟩$    c) $-|0⟩$    d) $-|1⟩$    e) $\frac{1}{\sqrt{2}}|0⟩+\frac{1}{\sqrt{2}}|1⟩$        

8. We have a qubit in state $|0⟩$. We apply the operators $X,H,X,H,X$ in order. What is the final state?

    a) $|0⟩$    b) $|1⟩$    c) $-|0⟩$    d) $-|1⟩$    e) $\frac{1}{\sqrt{2}}|0⟩-\frac{1}{\sqrt{2}}|1⟩$        

9. We apply a series of quantum operators to a single qubit that is in state $|0⟩$ at the beginning. If we observe the state $0$ at the end, which of the following combinations is not possible, where $M$ stands for a measurement and we apply the operators from the left to the right?

    a) $H,H,M$    b) $X,X,M$    c) $M,X,M,X,M$    d) $H,X,H,M$    e) $X,H,X,H,M$        

10. We have five qubits, say $q_0,\dots ,q_4$ initially in zero states. We apply $X$ operator to $q_0$ and $q_4$. For the rest of qubits, we apply either identity operator or $X$ operator. After making a measurement, we read the values from the qubits $q_0,\dots ,q_4$ as $b_0,\dots ,b_4$, respectively.

If $b=b_4\cdots b_0$ is a binary number, which of the following decimal numbers cannot be a value of $b$?

    a) $13$    b) $17$    c) $23$    d) $25$    e) $31$        

11. Hadamard operator $H=\left(\begin{array}{rr}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\end{array}\right)$ is a quantum operator and it preserves the length of any vector. Which one of the following operators is not a quantum operator? (Hint: Test each matrix with a few quantum vectors, e.g., $|0⟩$, $|1⟩$, $|+⟩$, $|-⟩$, etc.)

    a) $\left(\begin{array}{rr}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right)$    b) $\left(\begin{array}{rr}\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right)$    c) $\left(\begin{array}{rr}-\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right)$    d) $\left(\begin{array}{rr}-\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right)$    e) $\left(\begin{array}{rr}-\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right)$        

12. What is $XH$?

    a) $\left(\begin{array}{rr}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right)$    b) $\left(\begin{array}{rr}\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right)$    c) $\left(\begin{array}{rr}-\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right)$    d) $\left(\begin{array}{rr}-\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right)$    e) $\left(\begin{array}{rr}-\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right)$        

13. What is $(XH{)}^2|0⟩$?

    a) $|0⟩$    b) $|+⟩$    c) $|1⟩$    d) $|-⟩$    e) $-|0⟩$        

14. What is $(XH{)}^3|0⟩$?

    a) $|+⟩$    b) $-|+⟩$    c) $|1⟩$    d) $|-⟩$    e) $-|-⟩$        

15. What is $(XH{)}^5|0⟩$?

    a) $|+⟩$    b) $-|+⟩$    c) $|1⟩$    d) $|-⟩$    e) $-|-⟩$        

16. What is $(XH{)}^8|0⟩$?

    a) $|0⟩$    b) $|+⟩$    c) $|1⟩$    d) $|-⟩$    e) $-|0⟩$        

17. When a qubit is in the quantum state $|u⟩=\left(\begin{array}{r}\frac{3}{5}\\ -\frac{4}{5}\end{array}\right)$, Hadamard operator is applied: $|u^{\prime }⟩=H|u⟩$. What is the probability of being in state $|1⟩$ in the new quantum state $|u^{\prime }⟩$?

    a) $0.02$    b) $0.36$    c) $0.50$    d) $0.64$    e) $0.98$        

18. If $|u⟩=\left(\begin{array}{c}x\\ 3x\end{array}\right)$ is the quantum state of a qubit, what is the probability of being state $|0⟩$?

    a) $0.1$    b) $0.25$    c) $0.50$    d) $0.75$    e) $0.9$        

19. If $|u⟩=\left(\begin{array}{r}x\\ -2x\\ 2x\\ -x\end{array}\right)$ is the quantum state of a quantum system with four states, what is the probability of being in the state having amplitude $-2x$?

    a) $0$    b) $0.2$    c) $0.4$    d) $0.6$    e) $0.8$        

20. We have a qubit, and we have a counter with value 0.

Repeat 20 times:

• Set the state of qubit to $|0⟩$.
• Apply Hadamard operator.
• Make a measurement.
  • If state '0' is observed: the value of counter is increased by 2 and continue with the next iteration.
• Apply Hadamard operator.
• Make a measurement.
  • If state '1' is observed: the value of counter is decreased by 2 and continue with the next iteration.
• Apply Hadamard operator.
• Make a measurement.
  • If state '0' is observed: the value of counter is increased by 1
  • If state '1' is observed: the value of counter is decreased by 3

What is the expected value of counter at the end of the iterations?

    a) $-10$    b) $-5$    c) $0$    d) $5$    e) $10$