Quantum information Flashcards
Draw the Bloch sphere
Write a general state on the Bloch sphere
cos(o/2) |0> + e^iphi sin(o/2) |1>
Axes intercepts of Bloch sphere
x - |+>
y - |0> + i|1>
z - |0>
Why is it important that propagators are unitary (2)?
- They are reversible since they have inverses
- They are length preserving so represent rotations on the Bloch sphere
Write the density matrix of a general state
Pure state =
a density state that can be written as the outer product of a bra and ket
Write the Pauli matrices
Mixed state =
a weighted sum of pure states that cannot be expressed as a single pure state
Trace of a density matrix
1
General X roation gate
General Y rotaion gate
General Z rotation gate
Square root of X gate
1 -i
-i 1
(/sqrt(2))
Square root of Y gate
1 -1
1 1
\root2
Sqauare root of Z gate
1 0
0 i
/root2
How can H be created from 90* X and Z gates
H = 90Z 90X 90Z
How can X be created from H and Z
X = HZH
How can a general X rotation be created from a general Z rotation and vice-versa
Rx = H Rz H
Rz = H Rx H
Why can some gates be non-unitary
Because quantum evolution is unitary for iolated states - clearly this is not always the case
Rabi flopping =
cyclic flopping between two states in the presence of an oscillatory driving field
What is the condition for flopping to occur?
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)
rotating wave approximation =
can neglect terms that osciallate rapidly
Raman transition =
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.
How can Rabi flopping be used to implement a roation gate?
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
Hamiltonian of a spin 1/2 system in a magnetic field
H = 1/2 h_bar w_l
Where w_l is the larmor frequency
Derive an expression for a Hamiltonian transformed into the rotating frame
How can an arbitary x rotation be implemented using B field on a spin 1/2 system?
By applying the field for the correct length of time
Differnce between ordinary and extraordinary axes
Ordinary - refractive index independent on direction of polarisation
Extraordinary - refractive index dependent of direction of polarisation
Entangled state =
a state of a system of two or more qubits which cannot be writte as direct products of single qubit states
Maximally entangled state =
any state that can be converted to a Bell state by a unitary operation on a single qubit
When is a meausrement perfect?
When the axis of measurement is parallel to the state on the Bloch sphere
Propagator corrsponding to sigma_i
exp(-ix sigma_i) = cos(x) sigma_0 - isin(x) sigma_i
