prepared by Abuzer Yakaryilmaz (QLatvia)
Convention: The default direction of the rotations is counter-clockwise.
The rotation on the unit circle with angle $ \theta $ is denoted $ R(\theta) $. What is the matrix form of $ R(\theta) $?
(Hint: Apply each candidate matrix to states $ \ket{0} $ and $ \ket{1} $ to verify whether the result is the rotated state.)
Which of the following matrices represents the rotation with angle $ \frac{\pi}{6} $ on the unit circle?
If $ R(\theta) = \mymatrix{rr}{ -\sqrttwo & -\sqrttwo \\ \sqrttwo & -\sqrttwo } $, what is $ \theta $?
If $ R(\theta) = \mymatrix{rr}{ -\sqrttwo & \sqrttwo \\ -\sqrttwo & -\sqrttwo } $, what is $ \theta $?
If $ R(\theta) $ is applied to a qubit initially in state $ \ket{0} $ three times, what is the final state?
If $ R(-\theta) $ is applied to a qubit initially in state $ \ket{0} $ three times, what is the final state?
If $ R(\theta) $ is applied to a qubit initially in state $ \ket{1} $ twice, what is the final state?
The rotation operator $ R(\frac{3\pi}{7}) $ is applied to a qubit initially in state $ \ket{0} $ $n$ times. If the final state is $ \ket{0} $, which one of the followings can be a value of $ n $?
We have a qubit in state $ \ket{0} $. The rotations $ R(\frac{\pi}{3}) $ and $ R(\frac{\pi}{6}) $ are applied $ m $ and $ n $ times, respectively. If the final state is $ -\ket{1} $, what can be the values of $ (m,n) $?
We have a qubit in state $ \ket{0} $. The rotations $ R(\frac{\pi}{3}) $ and $ R(-\frac{\pi}{6}) $ are applied $ m $ and $ n $ times, respectively. If the final state is $ -\ket{1} $, what can be the values of $ (m,n) $?
The reflection on the unit circle having the line of reflection with angle $ \theta $ is denoted $ Ref(\theta) $. What is the matrix form of $ Ref(\theta) $?
(Hint: Apply each candidate matrix to the states $ \ket{0} $ and $ \ket{1} $ to verify whether the result is the reflected state.)
If $ Ref(\theta) = \hadamard $, what is $ \theta $?
(Hint: Apply each candidate matrix to the states $ \ket{0} $ and $ \ket{1} $ to verify whether the result is the reflected state.)
If $ Ref(\theta) = \X $, what is $ \theta $?
(Hint: Apply each candidate matrix to the states $ \ket{0} $ and $ \ket{1} $ to verify whether the result is the reflected state.)
If $ Ref(\theta) = \Z $, what is $ \theta $?
(Hint: Apply each candidate matrix to the states $ \ket{0} $ and $ \ket{1} $ to verify whether the result is the reflected state.)
What is the matrix form of the reflection having the line of reflection $ y=-x $?
Which of the followings is identical to $ \mymatrix{rr}{ -1 & 0 \\ 0 & 1 } $, where $ Z = \Z $ ?
(Hint: Test each candidate whether it maps the state $ \myvector{x \\ y} $ to the state $ \myvector{ -x \\ y } $.)
What is $ Ref(\theta) \cdot \myvector{ \cos \theta' \\ \sin \theta' } $?
Let $ \ket{u} $ is a quantum state on the unit circle with angle $ \theta' $. If we apply the operators $ Ref(\theta_1) $ and $ Ref(\theta_2) $ in order, what is the angle of the final state?
Which one of the following operators maps the state $ \myvector{\cos \theta \\ \sin \theta} $ to the state $ \myvector{\cos (- \theta) \\ \sin (- \theta)} $?
(Hint: Determine (i) whether $ \sin \theta = \sin (-\theta) $ or not and (ii) whether $ \cos \theta = \cos (-\theta) $ or not.)
Which one of the following operators is identical to $ Ref(\theta) $?
(Hint: Any arbitrary state, say $ \ket{u} $, on the unit circle is represented by its angle, say $ \theta' $. Find the angle of $ Ref(\theta) \ket{u} $ and compare it with the angle of each quantum state obtained by applying the candidate operators.)
Let $ \ket{u_1} = \myvector{ \cos \theta_1 \\ \sin \theta_1 } $ and $ \ket{u_2} = \myvector{ \cos \theta_2 \\ \sin \theta_2 } $ be two different quantum states, where $ \theta1, \theta_2 \in (0,\pi) $. If the probabilities of being in states $ \ket{0} $ for $ \ket{u_1} $ and $ \ket{u_2} $ are the same, what is the relation between $ \theta_1 $ and $ \theta_2 $?
Which one of the following pairs of quantum states is perfectly distinguishable?
Which one of the following pairs of quantum states is perfectly distinguishable?
Which one of the following pairs of quantum states cannot be distinguishable?
Which one of the following pairs of quantum states cannot be distinguishable?