Coursework focused on Chapter 2 of * Quantum Computing: A Gentle Introduction*.

#### Exercise 1

Let the direction of polaroid ’s preferred axis be given as a function of , , and suppose that the polaroids and remain horizontally and vertically polarized as in the experiment of Section 2.1.1 of Quantum Computing: A Gentle Introduction. What fraction of photons reach the

screen? Assume that each photon generated by the laser pointer has random polarization.

50% of photons emitted from the laser are blocked and 50% pass through polaroid with a state of . As polaroid preferred axis is already provided as a function of , we have:

which can be used to represent the fraction of photons that are permitted to pass through the second filter as:

Which simplifies down to:

in the state

Photons passing through polaroid C will be allowed to pass through when in a state, which we represent as:

in the state

Combined the resultant probability of photons passing through polaroid through is:

#### Exercise 2

Which pairs of expressions for quantum states represent the same state? For those pairs that represent different states, describe a measurement for which the probabilities of the two outcomes differ for the two states and give these probabilities.

**a.** and are the **same state**.

**b.** and are the **same state**.

**c.** and

and

**d.** and

and

**same state**

**e. ** and

and

**same state**

**f. ** and

and

**g. ** and

**same state**

**h. ** and

**same state**

**i. ** and

and

∴ and **same state**

**j. ** and

and

#### Exercise 3

Which states are superpositions with respect to the standard basis, and which are not? For each state that is a superposition, give a basis with respect to which it is not a superposition. Which of these are in states of superposition with regard to the Hadamard basis?

**a. **

This state is a superposition with regard to the standard basis , but not the Hadamard basis of because it’s a basis vector of the Hadamard basis.

**b. **

This state is not a superposition with regard to the standard basis, but is a superposition with regard to the Hadamard basis.

**c. **

This state is not a superposition with regard to the standard basis, but is a superposition with regard to the Hadamard basis.

**d. **

This is a superposition state, but is a superposition with regard to the Hadamard basis.

**e. **

This state is not a superposition with regard to the standard basis, but is a superposition with regard to the Hadamard basis.

**f. **

This state is a superposition with regard to the standard basis , but not the Hadamard basis of because it’s a basis vector of the Hadamard basis.

#### Exercise 4

For each pair consisting of a state and a measurement basis, describe the possible measurement outcomes and give the probability for each outcome.

**a. **,

**b. **,

**c. **

**d. **

**e. **

**f. **

**g. **

#### Exercise 5

**a. **Show that the surface of the Bloch sphere can be parametrized in terms of two real-valued parameters, the angles and illustrated in the figure above. The parametrization is in one-to-one correspondence with points on the sphere, and therefore single-qubit quantum states, in the range and except for the points corresponding to and .

Global Phase:

Normalization Condition:

∴

**b. **What are and for each of the states ?

∴