Coursework focused on Chapter 4 & 5 of * Quantum Computing: A Gentle Introduction*.

#### Exercise 4.1

Give the matrix, in the standard basis, for the following operators:

**a.**

**b.**

**c.**

**d.**

#### Exercise 4.2

Write the following operators in bra/ket notation:

**a.** The Hadamard operator:

**b.**

**c.**

**d.**

**e.**

#### Exercise 4.7

Using the projection operator formalism:

**a.** Compute the probability of each of the possible outcomes of measuring the first qubit of anÂ arbitrary two-qubit state in the Hadamard basis

AND

Probability of measuring

Probability of measuring

**b.** Compute the probability of each outcome for such a measurement on the state

and

Probability of measuring

Probability of measuring

#### Exercise 4.14

Give the outcomes and their probabilities for measurement of a general two-qubit state with respect to the Bell decomposition.

#### Exercise 5.4

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

#### Exercise 5.6

**a.** Show that is a rotation of about the y-axis of the Bloch sphere

IF AND

Therefore,

**b.** Show that is a rotation of about the z-axis of the Bloch sphere

IF AND

Therefore,