Quantum information

Question 1

Draw the Bloch sphere

Answer 1

Question 2

Write a general state on the Bloch sphere

Answer 2

cos(o/2) |0> + e^iphi sin(o/2) |1>

Question 3

Axes intercepts of Bloch sphere

Answer 3

x - |+>

y - |0> + i|1>

z - |0>

Question 4

Why is it important that propagators are unitary (2)?

Answer 4

1. They are reversible since they have inverses
2. They are length preserving so represent rotations on the Bloch sphere

Question 5

Write the density matrix of a general state

Answer 5

Question 6

Pure state =

Answer 6

a density state that can be written as the outer product of a bra and ket

Question 7

Write the Pauli matrices

Answer 7

Question 8

Mixed state =

Answer 8

a weighted sum of pure states that cannot be expressed as a single pure state

Question 9

Trace of a density matrix

Answer 9

1

Question 10

General X roation gate

Answer 10

Question 11

General Y rotaion gate

Answer 11

Question 12

General Z rotation gate

Answer 12

Question 13

Square root of X gate

Answer 13

1 -i

-i 1

(/sqrt(2))

Question 14

Square root of Y gate

Answer 14

1 -1

1 1

\root2

Question 15

Sqauare root of Z gate

Answer 15

1 0

0 i

/root2

Question 16

How can H be created from 90* X and Z gates

Answer 16

H = 90Z 90X 90Z

Question 17

How can X be created from H and Z

Answer 17

X = HZH

Question 18

How can a general X rotation be created from a general Z rotation and vice-versa

Answer 18

Rx = H Rz H

Rz = H Rx H

Question 19

Why can some gates be non-unitary

Answer 19

Because quantum evolution is unitary for iolated states - clearly this is not always the case

Question 20

Rabi flopping =

Answer 20

cyclic flopping between two states in the presence of an oscillatory driving field

Question 21

What is the condition for flopping to occur?

Answer 21

Must have V = non zero ie the perturbation must link the two levels. For it to be efficient V >> w_0 (frequency of laser field)

Question 22

rotating wave approximation =

Answer 22

can neglect terms that osciallate rapidly

Question 23

Raman transition =

Answer 23

where populations are driven between two levels between which direct transitions are forbiden by using a third higher level. To acheive Rabi flopping on a two level system, two laser beams corresponding to the two levels are detuned from the desired transition to the third level by the same amount.

Question 24

How can Rabi flopping be used to implement a roation gate?

Answer 24

The Hamiltonian is

0 hV/2

hV/2 0

Which corresponds to an X rotation where theta = Vt

So to perform a rotation choose the correct time period

Question 25

Hamiltonian of a spin 1/2 system in a magnetic field

Answer 25

H = 1/2 h_bar w_l

Where w_l is the larmor frequency

Question 26

Derive an expression for a Hamiltonian transformed into the rotating frame

Answer 26

Question 27

How can an arbitary x rotation be implemented using B field on a spin 1/2 system?

Answer 27

By applying the field for the correct length of time

Question 28

Differnce between ordinary and extraordinary axes

Answer 28

Ordinary - refractive index independent on direction of polarisation

Extraordinary - refractive index dependent of direction of polarisation

Question 29

Entangled state =

Answer 29

a state of a system of two or more qubits which cannot be writte as direct products of single qubit states

Question 30

Maximally entangled state =

Answer 30

any state that can be converted to a Bell state by a unitary operation on a single qubit

Question 31

When is a meausrement perfect?

Answer 31

When the axis of measurement is parallel to the state on the Bloch sphere

Question 32

Propagator corrsponding to sigma_i

Answer 32

exp(-ix sigma_i) = cos(x) sigma_0 - isin(x) sigma_i