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Flashcards in QIC Deck (46):
1

Limits of Classical Computing (2)

1) Moore's law - double no. transistors on chip in 18 months, will reach size of atom by 2020
2) Thermodynamics - logic gates have 2ips and 1op - loss of information, change of entropy, gives off heat which must be dissipated

2

General Qubit State
N Qubit state

|phi> = a|0>+b|1>
Isomorphic to spin 1/2 system
|phi> = a1|00..0> + a2|00..1>.. 2^N coefficients - exponential scaling gives quantum power

3

How do you achieve single-qubit rotation?

Resonant EM field

4

What happens to 2 level system subject subject to resonant excitation?

Rabi oscillations

5

Physical explanation of origin of population oscillations

Populations oscillate with Rabi oscillations, cycles of absorption and stimulated emission

6

2 Kinds of rotation of state on Bloch Sphere

1) Free evolution - precesion around z axis
2) Interatction with an external field - takes qubits from |0> or |1> to equitorial plane

7

Bloch Sphere positions

+z : |0>
-z : |1>
(p = 1/root2)
+x : p(|0>+|1>)
-x : p(|0>-|1>)
+y : p(|0>+i|1>)
-y : p(|0>-i|1>)

8

theta and phi L

thetaL = extent of rotation, determined by strength and duration of pulse
PhiL gives axis about which rotation occurs, determined by phase of external field

9

pi/2 pulse, pi pulse, 2pi pulse, 4pi pulse

pi/2 - takes you from |0> to -x or +x
pi - change from 0 to 1 and vice versa
2pi - 0 back to top of sphere, global sign not represented
4pi - recovers original state of spin 1/2 system

10

2 Applications of Ramsey interferometry

1)Atomic clocks
2) Measurement of fundamental constants

11

Qubit A has state vector |phi>a in vector space Ha and Qubit B has same in B, 2 qubit state |phi>ab lives in?

Ha tensor product Hb

12

What is density operator rho?

outer product
rho = |psi>

13

Difference between product and entangled state, example of each

Product state |psi>ABC has 6 = 2N coefficients, but general state has 8 = 2^N - can't make all states, others are entangled, can't be written as prouct of 6-qubit state
eg 1/root2(|00>+|11>)

14

How do you use density matrix formulism to calculate expectation value of operator M

= Tr(rho M) = Tr(M rho)

15

Classical correlations

Socks - measuring string of L and R, measurements are correlated

16

EPR Paradox

It is tempting to think that qubits A and B have well defined states after they interact but this does not produce the same correlations as QM result

17

Bell states

p = 1/root2
|phi+-> = p(|01>+-|10>)
|psi+-> = p(|00>+-|11>)

18

How to engineer entangled state from product state with controlled collision

Start with qubits in neighbouring boxes, both state 0, apply pi/2 pulse to both - positive x direction. Displace 0 component of A towards 1 component of B - state evolves and gives entangled state.

19

CNOT gate

conditional Ramsey interferometer, qubits labelled target and control
2 i/p 2 o/p not fighting thermo, reversible - know what went in

20

Truth table for CNOT

IN OUT
00 00
01 01
10 11
11 10
If control is 1, target changes, pi phase change

21

CNOT universal

Network of CNOT gates and single qubit rotation is sufficient for performing arbitrary computation

22

5 DiVincenzo criteria

1)Scalable system with well characterized qubits
2) Initialization into well defined state
3) Long decoherance times
4) Universal set of quantum gates
5) Efficient procedure for measuring state of qubits at end of calculation

23

What might archtecture of QC look like?

Input -> quantum logic gates -> output
000, 001 --> Function of 0's and ones

24

Choices of qubits

Photons - good for conveying info but not storing (travel at c)
NMR - no scalability
Quantum Dots - CM
Superconductiong Josephson Junction - CM
Laser cooled atoms

25

Laser cooling hinges on 3 effects

1) Radiation Pressure
2) Doppler effect
3) Resonance

26

Momentum photon in plane wave

p = hbar k
Atom that absorbs photon recoils to conserve momentum

27

How is it possible to use lasers to cool atoms?

Atoms absorb photons and recoil, typical change of speed 3cms-1, absorb photons to slow atoms, need closed system
Detuning
Resonance, atom going away from beam shifts left, right shifts closer to resonance, net force opposes motion - friction

28

If detuning of laser from resonance for stationary atom is delta, for atom with vel v?

delta - k.v

29

Atomic absorption line

Power broadened lorentzian with scattering rate R (in notes)

30

How do you use optical molasses in 3D

3 pairs of counterpropagating beams which are mutually orthogonal

31

What happens in optical molasses?

The atoms get into equilibrium when laser cooling balances momentum diffusion

32

Lowest achievable temperature

kbTdoppler = 1/2 hbar gamma

33

Trapping atoms

Neutral atoms can be trapped with dipole potentials

34

Stark shift - lightshift

Look at graphs
Light shift interpreted as ac stark shift

35

Origin of Dipole Force

E induces dipole d in atom
If wL>w0 - d.E out of phase by pi -d.E is positive

36

Geometries for strong dipole force

1) counterpropagating beams - can extend to 3D with more beams, optical lattice.
2)Focussed laser beam - steep gradient: large force

37

Trapping atoms delta

Trap atoms at intensity maxima if delta is negative
Trap at minima if delta is positive

38

Principles of laser cooling and importance of each

i) Radiation pressure - provides momentum transfer
ii) Doppler effect - ensures force opposes motion
iii) resonance - detuning laser frequency in vicinity of resonance provides strong frictional force

39

KE group of atoms

ke = 3/2 N k_b T
approx 100 micro K so turns out to be change of ke of 3/2 N kb T

40

Where does energy go Doppler?

Radiated light carries away energy as average energy of emitted photon larger than energy used for laser cooling

41

CNOT, inverse CNOT, swap

1000
0100
0001
0010

1000
0001
0010
0100

1000
0010
0100
0001

42

R(pi/2, pi/2)

1 -1
1 1

43

Eigenenergies

+- hbar/2 (omega^2 + delta^2)^1/2
E1,2 = +- hbar/2 (Delta/2 + omega^2/4delta)

first term E difference between atom and light, second term is light shift or ac stark shift

44

Scattering rate

As more atoms are scattered, atom accelerates and is doppler shifted off resonance, rate decreases
Force is R x energy (hbar k)

45

How to check eigenvalue

Want to know if k is eigenvalue of A
Ak = lamda k

46

Principle of Ramsey Interferometry

Start with atom in 0
Apply pi/2 pulse with resonant em field, creates linear superposition of 0 and 1.
Paths acquire different phases
Second pulse recombines paths, output 0 or 1 with prob cos^2 phi and sin^2 phi respectively.