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Week 4: SINGLE QUBIT QUANTUM SYSTEMS | ECE 802-730

# Week 4: SINGLE QUBIT QUANTUM SYSTEMS | ECE 802-730

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 ?