prepared by Abuzer Yakaryilmaz (QLatvia)

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


If $ \ket{u} = \myvector{ \frac{1}{2} \\ x \\ y} \in \mathbb{R}^3 $ is a quantum state, which one of the following equations cannot be possible?


We have a three state quantum system. If the system is in the quantum state $ \ket{u} = \myvector{ \frac{1}{3}-\frac{1}{\sqrt{3}} \\ \frac{1}{3}+\frac{1}{\sqrt{3}} \\ x } \in \mathbb{R}^3 $, what is the probability of being in the third state?


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


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


What is $ H^7 $?


We have a qubit in state $ \ket{0} $. We apply the operators $ H, X, X, H, X $ in order, where $ X $ is NOT operator. What is the final state?


We have a qubit in state $ \ket{0} $. We apply the operators $ X, H, X, H, X $ in order. What is the final state?


We apply a series of quantum operators to a single qubit that is in state $ \ket{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?


We have five qubits, say $ q_0,\ldots,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,\ldots,q_4 $ as $ b_0,\ldots,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 $?


Hadamard operator $ H = \mymatrix{rr}{ \sqrttwo & \sqrttwo \\ \sqrttwo & -\sqrttwo } $ 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., $ \ket{0} $, $ \ket{1} $, $ \ket{+} $, $ \ket{-} $, etc.)


What is $ XH $?


What is $ (XH)^2\ket{0} $?


What is $ (XH)^3\ket{0} $?


What is $ (XH)^5\ket{0} $?


What is $ (XH)^8\ket{0} $?


When a qubit is in the quantum state $ \ket{u} = \myrvector{\frac{3}{5} \\ -\frac{4}{5}} $, Hadamard operator is applied: $ \ket{u'} = H \ket{u} $. What is the probability of being in state $ \ket{1} $ in the new quantum state $ \ket{u'} $?


If $ \ket{u} = \myvector{x \\ 3x} $ is the quantum state of a qubit, what is the probability of being state $ \ket{0} $?


If $ \ket{u} = \myrvector{x \\ -2x \\ 2x \\ -x } $ is the quantum state of a quantum system with four states, what is the probability of being in the state having amplitude $ -2x $?


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

Repeat 20 times: What is the expected value of counter at the end of the iterations?