Quantum Mechanics and Quantum Computation

1
Quantum MechanicsandQuantum Computation

• Sarah Brandsen

2

• If quantum mechanics hasnt profoundly shocked
  you, you havent understood it yet.
• -Niels Bohr

3
Outline

• What is Quantum Mechanics?
• Phenomena
• Quantization
• Wavefunctions and Schrodingers Cat
• Quantum Computing
• What is Computing?
• Why might Quantum Computers be useful?
• National Security and Quantum Computers
• Quantum Key Distribution (QKD)
• Polarized Photons, Heisenberg Uncertainty,
  No-Cloning Theorem
• Is QKD a perfect form of encryption?
• When will we have quantum computers?

4
What is Quantum Mechanics, really?

• Our attempt to understand the universe on
  extremely small (nanoscopic) scales
• This leads to strange and unintuitive results-
• Momentum and energy can be quantized
• Light and matter can both exhibit wave and
  particle behaviour

5
Most Famous Observation in Physics- Schrodingers
Cat

• In fact, the mere act of opening the box will
  determine the state of the cat, although in this
  case there were three determinate states the cat
  could be in these being Alive, Dead, and Bloody
  Furious.
• - Terry Pratchett

6
Set-up for Schrodingers Cat

• Place cat in box with a purely random radioactive
  source that has 50 of going off (killing the
  cat) within an hour
• At the end of the hour, right before you open the
  box, the cats state is completely unknown
• Therefore, the cat is considered to be both alive
  and dead at the same time until it is observed.

7
Wavefunction

• The wavefunction represents all possible outcomes
  and their probabilities

Square this to find probability of cat being
alive 50
State of cat being alive
8
Measurement and Zombie Cats

• If cats can be in superpositions of dead and
  alive, why dont we see zombie cats?
• According to a standard interpretation,
  measurement collapses the wavefunction into a
  single state!

9
Quantum Computing- Active Research based on
Quantum Mechanics
10
What is computing?

• Goal-oriented activity
• Store information in bits (0 or 1) and processes
  the information similarly to computation by hand
• Very effective for many tasks, but has time
  limitations for other tasks

11
Example- Computer Chess Game

• Input- Your move of the chess piece
• Process- Based on your move, the computer
  calculates a response
• Output- Computer makes a move, which appears on
  your screen

12
What is quantum computing?

• Exploiting the strange phenomena in quantum
  mechanics to process information
• Quantum computers can be in a superposition of
  several states, so you can process information in
  parallel

13
Bit vs. Qubit

• Superposition of classical states, so you can
  contain as many inputs as you want and process
  in parallel!

14
Measurement problems?

• Quantum computing is probabilistic- we could run
  exactly the same program twice and see different
  results
• But you can repeatedly initialize, run, and
  measure

15
Thought Experiment

• Impossible task-- find a small X that has been
  marked on one book in the Library of Congress in
  less than five minutes
• But what if there were several million parallel
  Library of Congress and each one had a parallel
  self? If you could coordinate with all of the
  parallel selves, you might be successful.

16
Why Quantum Computing is a Big Deal

• Can solve enormous tasks in a reasonable amount
  of time finding a needle in a haystack
• True random number generation
• Implications for artificial intelligence?
• National security

17
More about National Security

• Our daily lives depend on encryption
• Encodes message so that they are useless to
  anyone who doesnt know the cipher
• Prevents eavesdroppers on mobile phone
  conversations, protects information being
  transferred over the network, etc.
• Symmetric key scheme

18
Another kind of Encryption

• Banking, medical records, business records, and
  government records often use RSA encryption tools
• This relies on the difficulty of breaking a very
  large number into two prime factors

Factors of a very large number!
19
Safe For now

• In 2009, computer scientists were able to crack
  the primes for a 768-bit number over the span of
  two years and using hundreds of computers
• Many encryption keys are 1024 bits, so it would
  even take 1000 times longer to crack!
• However, quantum computers could use a process
  called Shors algorithm to crack very large
  encryption keys

20
Quantum Key Distribution (QKD)

• QKD depends on three properties
• Photon Polarization
• Heisenberg uncertainty principle
• No-cloning theorem

21
Photon Polarization

• Unpolarized (normal) photons can spin in all of
  their directions at once
• Can be polarized by passing through a filter
• Cant be re-polarized without losing information

22
Uncertainty Principle

• Some kinds of knowledge are fundamentally
  inaccessible
• For example, the more precisely you measure a
  particles position the less precisely you can
  know its momentum.
• It is also impossible to know a photons spin in
  two directions at once!

23
Quantum No-Cloning Theorem

• Suppose you can create multiple identical
  properties of a quantum state
• Then you can measure position in one state and
  momentum in another
• This violates the Heisenberg uncertainty
  principle!

24
BB84 Protocol

• Alice chooses a random polarization and records
  her result
• Bob also chooses a random polarization and
  records his result

25
BB84 Protocol

• At the end, Alice can read off what polarizing
  filters she used and they can keep sets that
  agree
• This will give them a private code!

26
What if Eve was eavesdropping?

• Eve would have to randomly measure the photon to
  obtain information.
• Half of the time she will choose the wrong
  filter. Then if Bob chooses the right filter, his
  result will be random and might be different from
  Alices
• Bob and Alice can compare a small portion of
  their code over the classical channel and see
  whether Eve was eavesdropping

27
Problems with Quantum Encryption

• Noisy and imperfect equipment might make Bob and
  Alices measurements disagree
• Imperfect photon source helps Eve eavesdrop

28
Problems with Quantum Computation

• Quantum systems are very delicate
• Decoherence (interactions with the environment)
  can quickly collapse the wavefunction

29
When will quantum computers be made?

• 1998- First three qubit computer
• 2012- Decoherence suppressed for 2 seconds at
  room temperature
• 2013- Decoherence suppressed for 39 minutes at
  room temperature in an ensemble of 3 billion
  qubits

30
Thank you for Listening
31
Tunneling

• Consider a ball rolling off into a valley, bound
  on the other side by a higher hill. Classic
  physics states that the ball cannot have
  sufficient energy to roll over from the other
  side of the higher hill unless an external
  velocity is imparted on this ball, but according
  to quantum tunneling, there is the minute
  possibility that this ball will turn out to be
  that one coincidence out of a million and DIG
  THROUGH the hill instead of rolling over.

32
Black Hole Information Paradox

• The reason that this lost information is such a
  major issue for quantum mechanics once again ties
  into thermodynamics. In quantum mechanics,
  information is related to the thermodynamic
  concept of order. If information is lost, then
  order is lost meaning that entropy (disorder)
  is increased. This means that the black holes
  would begin generating heat, rising up to
  billions of billions of degrees in mere moments.
  Though Leonard Susskind and others realized this
  in the mid-1980s, they couldnt find the flaws in
  Hawkings reasoning that would prove him wrong.
• he redid some of his earlier calculations and
  found that it was possible that, as an object
  fell into a black hole, it would disturb the
  black holes radiation field. The information
  about the object could seep out, though probably
  in mangled form, through the fluctuations in this
  field.
• it may be possible that all the information
  within the black hole is also encoded in some
  form on the surface area of the black hole.

33
Uncertainty Principle in Pictures
More Precise Position, Less Precise Momentum
Precisely determined momentum, unknown position
34
Quantum Entanglement- Spooky Action At A
Distance

• ?Particles can be connected, even across great
  distances!
• ?Example- laser beam fired through specific
  crystal can cause individual photons to split
  into pairs of entangled photons
• When you measure the spin of particle A, you
  determine the spin of particle B virtually
  instantaneously!

35
Alternative View- Many Worlds Approach

• Wavefunction doesnt magically collapse upon
  observation!
• Instead, we start with a superposition
• After measuring, the superposition is still
  there but the different branches can no longer
  interact with each other
• This is due to decoherence

36
Many-Worlds Interpretation

• Schrodingers cat is alive in one world and dead
  in another
• When you open the box you can say which world
  you are in. You are either in world-1 or world-2,
  but both the worlds exist.

37
Phenomena 1- Quantization

• In 1900 Max Planck started a revolution in
  physics without realizing it
• Blackbody- an object that perfectly absorbs and
  emits radiation
• Experimental data could be fit by assuming that
  energy was quantized!

38
More Quantization
Electrons restricted to specific energy levels
39
More Quantization

• Energy
• Momentum/Velocity

40
What is light?

• Debate goes all the way back to Aristotle and
  Democritus
• Lots of evidence for light to be a wave

41
Light as a Particle

• More intense light doesnt lead to more intense
  radiation
• 1905- Einstein hypothesized that light may be
  composed particle-like pieces (photons)
• Existence of photons later confirmed (1923)

42
What about matter?

• Familiar with definition of matter as a
  collection of particles (for example, a
  baseball).
• We have never observed baseballs diffracting
  around corners

43
Double Slit Experiment

• Light produces interference patterns when passed
  through a slit

44
Electrons can also act like a wave

• Interference builds up over time

45
Recap

• Momentum, energy can be quantized
• Light can act like a wave and a particle
• Matter can act like a group of particles and a
  wave

46
Even Stranger- Double Slit Experiment Revisited

• If particle detectors are positioned at the slit,
  the interference pattern will disappear
• We cannot observe something acting as a particle
  and wave at once

47
Quantum Mechanics in Pop Culture
New Age Religion
Obscure Math
Image Source Church of the Cosmos
We are immortal timeless beings of light in an
infinitely expanding omniverse filled with
intelligent life.