Exercises are from QUANTUM COMPUTING: A GENTLE INTRODUCTION, by Eleanor Rieffel and Wolfgang Polak, published by The MIT Press.

$\def\abs#1{|#1}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 5.1}$

Show that any linear transformation $U$ that takes unit vectors to unit vectors preserves orthogonality: if subspaces $S_1$ and $S_2$ are orthogonal, then so are $U S_1$ and $U S_2$.

$\mathbf{Exercise\ 5.2}$

For which sets of states is there a cloning operator? If the set has a cloning operator, give the operator. If not, explain your reasoning.

a) $\{ \ket 0, \ket 1 \}$,

b) $\{ \ket +, \ket - \}$,

c) $\{ \ket 0, \ket 1, \ket +, \ket - \}$,

d) $\{ \ket 0\ket +, \ket 0\ket -, \ket 1\ket +, \ket 1\ket - \}$,

e) $\{ a\ket 0 + b\ket 1 \}$, where $|a|^2 + |b|^2 = 1$.

$\mathbf{Exercise\ 5.3}$

Suppose Eve attacks the BB84 quantum key distribution of Section 2.4 as follows. For each qubit she intercepts, she prepares a second qubit in state $\ket 0$, applies a $C_{not}$ from the transmitted qubit to her prepared qubit, sends the first qubit on to Bob, and measures her qubit. How much information can she gain, on average, in this way? What is the probability that she is detected by Alice and Bob when they compare $s$ bits? How do these quantities compare to those of the direct measure-and-transmit strategy discussed in Section 2.4?

$\mathbf{Exercise\ 5.4}$

Prove that the following are decompositions for some of the standard gates.

(1)$\mathbf{Exercise\ 5.5}$

A vector $\ket\psi$ is *stabilized* by an operator $U$ if $U\ket\psi = \ket\psi$. Find the set of vectors stabilized by

a) the Pauli operator $X$,

b) the Pauli operator $Y$,

c) the Pauli operator $Z$,

d) $X\otimes X$,

e) $Z\otimes X$,

f) $C_{not}$.

$\mathbf{Exercise\ 5.6}$

b) Show that $R(\alpha)$ is a rotation of $2\alpha$ about the $y$-axis of the Bloch sphere.

b) Show that $T(\beta)$ is a rotation of $2\beta$ about the $z$-axis of the Bloch sphere.

c) Find a family of single-qubit transformations that correspond to rotations of $2\gamma$ about the $x$-axis.

$\mathbf{Exercise\ 5.7}$

Show that the Pauli operators form a basis for all linear operators on a two-dimensional space.

$\mathbf{Exercise\ 5.8}$

What measurement does the operator $\i Y$ describe?

$\mathbf{Exercise\ 5.9}$

How can the circuit

be used to measure the qubits $b_0$ and $b_1$ for equality without learning anything else about the state of $b_0$ and $b_1$? (Hint: you are free to chose any initial state on the register consisting of qubits $a_0$ and $a_1$.)

$\mathbf{Exercise\ 5.10}$

An $n$-qubit cat state is the state $\frac{1}{\sqrt 2}(\ket{00\dots 0} + \ket{11\dots 1}$. Design a circuit which, upon input of $\ket{00\dots 0}$, constructs a cat state.

$\mathbf{Exercise\ 5.11}$

Let $\ket{W_n} = \frac{1}{\sqrt{n}} (\ket{0\dots 001} + \ket{0\dots 010} + \ket{0\dots 100} + \cdots + \ket{1\dots 000}).$

Design a circuit which, upon input of $\ket{00\dots 0}$, constructs $\ket{W_n}$.

$\mathbf{Exercise\ 5.12}$

Design a circuit that constructs the Hardy state $\frac{1}{\sqrt{12}} (3\ket{00} + \ket{01} + \ket{10} + \ket{11}).$

$\mathbf{Exercise\ 5.13}$

Show that the swap circuit of section 5.2.4 does indeed swap two single-qubit values in that it sends $\ket\psi\ket\phi$ to $\ket\phi\ket\psi$ for all single-qubit states $\ket\psi$ and $\ket\phi$.

$\mathbf{Exercise\ 5.14}$

Show how to implment the Toffoli gate $\bigwedge_2 X$ in terms of single-qubit and $C_{not}$ gates.

$\mathbf{Exercise\ 5.15}$

Design a circuit that determines if two single qubits are in the same quantum state. The circuit may include an ancilla qubit to be measured. The measurement should give a positive answer if the two-qubit states are identical, a negative answer if the two-qubit states are orthogonal, and be more likely to give a positive answer the closer the states are to being identical.

$\mathbf{Exercise\ 5.16}$

Design a circuit that permutes the values of three qubits in that it sends $\ket\psi\ket\phi\ket\eta$ to $\ket\phi\ket\eta\ket\psi$ for all single-qubit states $\ket\psi$, $\ket\phi$, and $\ket\eta$.

$\mathbf{Exercise\ 5.17}$

Compare the effect of the following two circuits

$\mathbf{Exercise\ 5.18}$

Show that for any finite set of gates there must exist unitary transformations that cannot be realized as a sequence of transformations chosen from this set.

$\mathbf{Exercise\ 5.19}$

Let $R$ be an irrational rotation about some axis of a sphere. Show that for any other rotation $R'$ about the same axis and for any desired level of approximation $2^{-d}$ there is some power of $R$ that approximates $R'$ to the desired level of accuracy.

$\mathbf{Exercise\ 5.20}$

Show that the set of rotations about any two distinct axes of the Bloch sphere generate all single-qubit transformations (up to global phase).

$\mathbf{Exercise\ 5.21}$

a) In the Euclidean plane, show that a rotation of angle $\theta$ may be achieved by composing two reflections.

b) Use part a) to show that a clockwise rotation of angle $\theta$ about a point $P$ followed by a clockwise rotation of angle $\phi$ about a point $Q$ results in a clockwise rotation of angle $\theta + \phi$ around the point $R$, where $R$ is the intersection point of the two rays, one through $P$ at angle $\theta/2$ from the line between $P$ and $Q$ and the other through point $Q$ at an angle of $\phi/2$ from the line between $P$ and $Q$.

c) Show that the product of any two rational rotations of the Euclidean plane is also rational.

d) On a sphere of radius $1$, a triangle with angles $\theta$, $\phi$, and $\eta$ has area $\theta + \phi + \eta$ (where $\theta$, $\phi$, and $\eta$ are in radians).

Use this fact to describe the result of rotating clockwise by angle $\theta$ around a point $P$ followed by rotating clockwise by angle $\phi$ around a point $Q$ in terms of the area of a triangle.

e) Prove that on the sphere the product of two rational rotations may be an irrational rotation.

$\mathbf{Exercise\ 5.22}$

a) Show that the gates $H$, $P_{\pi/2}$ and $P_{\pi/4}$ are all (up to global phase) rational rotations of the Bloch sphere. Give the axis of rotation and the angle of rotation for each of these gates, and also the gate $S = HP_{\pi/4}H$.

b) Show that the transformation $V = P_{\pi/4}S$ is an irrational rotation of the Bloch sphere.