The Mathematics of Star Trek

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The Mathematics of Star Trek

• Lecture 12 Quantum Computing

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Topics

• Security of RSA
• Thomas Youngs Light Experiment
• A Modern Version of Youngs Experiment
• Superposition
• Many-Worlds
• The Quantum Computer
• Applications of Quantum Computing
• Drawbacks to Quantum Computing

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Thomas Youngs Double-Slit Light Experiment

• Imagine two ducks swimming alongside each other
  in a pond.
• As each duck passes through the water, a trail of
  ripples will form behind the duck.
• The two sets of ripples fan out and interact -
  canceling out when a peak meets a trough and
  forming a higher peak when two peaks (or two
  troughs) meet.

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Thomas Youngs Double-Slit Light Experiment
(cont.)

• Starting in 1799, English physician and physicist
  Thomas Young (1773-1829) performed a series of
  experiments with light, including one in which a
  partition with two narrow vertical slits is
  placed between a light source and a screen.
• Young expected that there would be two bright
  stripes on the screen.
• Instead, he found that the light fanned out from
  the two slits and formed a pattern of several
  light and dark stripes on the screen.
• Handout of Youngs Experiment

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Thomas Youngs Double-Slit Light Experiment
(cont.)

• Assuming that light was a form of a wave, Young
  concluded that the light coming out of each slit
  was behaving like the ripples in the water behind
  the ducks.
• The dark and light stripes were caused by the
  same sort of interactions as the ripples in the
  water.
• Light stripes were caused by two peaks or two
  troughs of the light waves interacting.
• Dark stripes were caused by the interaction of a
  trough and a peak of the light waves.

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Thomas Youngs Double-Slit Light Experiment
(cont.)

• We now know that light does act like a wave (or a
  particle), but at the time of Youngs experiment,
  this was not well known.
• Young published his ideas on the nature of light
  in the classic paper The Undulatory Theory of
  Light.

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A Modern Version of Youngs Experiment

• We now know that light can be thought of a wave
  or made up of particles, called photons.
• Modern technology allows us to reproduce Youngs
  experiment with a light source capable of
  emitting single photons of light, at rates such
  as one photon per minute.
• As each photon travels towards the partition, it
  may pass through one of the slits.

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A Modern Version of Youngs Experiment (cont.)

• For this experiment, we use a screen made up of
  special photo detectors that can record each
  photon that makes it through the partition.
• Over a period of several hours, we will get an
  overall picture of where the photons are hitting
  the screen.
• Since individual photons are passing through the
  slits, we wouldnt expect to see the same striped
  interference pattern as we do for a regular light
  bulb.
• Handout of Modern Version of Youngs Experiment.

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A Modern Version of Youngs Experiment (cont.)

• Amazingly, we see the same pattern of light and
  dark strips as for Youngs original experiment,
  which means the individual photons are somehow
  interacting!
• This weird result defies common sense and there
  is no way to explain what is going on in terms of
  classical physics.
• Since photons are very small particles, we can
  try to use the ideas of quantum mechanics to
  explain what we see.

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A Modern Version of Youngs Experiment (cont.)

• It turns out that even experts in quantum
  mechanics cannot agree on what is happening!
• Right now there are two competing theories that
  are used to explain what is happening in the
  modern version of Youngs experiment.

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Superposition

• The first way to explain what is going on is via
  superposition.
• First of all, we only know two things for certain
  about an individual photon
• It leaves the light source.
• It strikes the screen.
• Everything else is a mystery, including if the
  photon passed through the left slit or the right
  slit.
• Since the exact path of the photon is unknown, we
  assume it passes through both slits
  simultaneously, which would allow the photon to
  interfere with itself, creating the pattern we
  see on the screen!

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Superposition (cont.)

• Here is how superposition works
• Each photon has two possible slits to pass
  through - left and right.
• We call each possibility a state and since we
  dont know which state the photon is in, we say
  it is in a superposition of states.
• One way to understand the idea of superposition
  is via a famous example suggested by Erwin
  Schrödinger (1887-1961)!

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Superposition (cont.)

• Suppose we have a (living) cat, a box, and a vial
  of cyanide.
• There are two possible states for the cat - dead
  or alive.
• Initially, the cat is in one of the two possible
  states, namely alive.
• Put the cat and vial of cyanide in the box and
  close the lid.
• Until we open the lid, we cannot see or measure
  the state of the cat.
• Quantum theory says the cat is in a superposition
  of two states - it is both dead and alive.
• Superposition occurs when we lose sight of an
  object and is a way of describing a period of
  ambiguity.
• Once we open the box, and look at the cat,
  superposition disappears and the cat is forced
  into one of its possible states.

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Many-Worlds

• The other way to describe what is going on with
  the modern Youngs experiment is via the
  many-worlds interpretation!
• Once the photon leaves the light source, since it
  has two possible slits to pass through, the
  universe splits into two universes.
• In one universe, the photon goes through the left
  slit.
• In the other universe, the photon goes through
  the right slit!
• These two universes somehow interfere with each
  other and produce the striped pattern of light
  and dark stripes.

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Many-Worlds (cont.)

• In the many-worlds theory, any time an object has
  the potential to enter one of several possible
  states, the universe will split into many
  universes, one for each potential state.
• The huge number of universes produced is called
  the multiverse.
• In the Star Trek Original Series episode Mirror,
  Mirror, we see an example of this phenomenon -
  there are two different universes - one where the
  Federation is good and another where the
  Federation is bad.

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Uses of Quantum Mechanics in the World Today

• Although strange and counterintuitive, many
  phenomena in the world owe their understanding or
  existence to quantum theory!
• Examples include
• Describing the modern version of Youngs
  experiment.
• Calculating consequences of nuclear reactions in
  power stations.
• Explaining how DNA works.
• Understanding how stars such as our Sun work.
• Designing lasers for CD players.

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The Quantum Computer

• Another consequence of quantum mechanics is the
  possibility of a quantum computer!
• In 1985, British physicist David Deutsch
  published a paper outlining how a computer might
  work according to the laws of quantum mechanics
  instead of classical physics.
• Such a computer would have to work at the level
  of fundamental particles for the quantum effects
  to manifest themselves.

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The Quantum Computer (cont.)

• So how would a quantum computer differ from a
  classical computer (i.e. the kind we use right
  now)?
• Suppose we have two versions of a question, say
  version 1 and version 2.
• To answer the question with a classical computer,
  we would have to perform the following sequence
  of operations
• Input version 1 and wait.
• Get an answer.
• Input version 2 and wait.
• Get and answer.

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The Quantum Computer (cont.)

• For a quantum computer, we do the following
• Combine the two questions as a superposition of
  two states, one for each question.
• Input this superposition to the computer, which
  causes the computer to enter a superposition of
  two states, one for each question, and wait.
• Get the answer to both questions at the same time!

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The Quantum Computer (cont.)

• As an illustration of how powerful a quantum
  computer could be, suppose we wish to answer the
  question Find the smallest positive integer
  whose square and cube use up all the digits 0-9
  once and only once.
• For a classical computer, wed need to do the
  following
• 1 12 1 and 13 1 NO
• 2 22 4 and 23 8 NO
• 3 32 9 and 33 27 NO
• 69 692 4,761 and 693 328,509 YES
• If each computation took one second, it would
  take 69 seconds to arrive at this answer.
• For a quantum computer, instead of testing one
  integer at a time, we could test many at once via
  superposition of states (or computation in many
  universes)!
• Thus, assuming one second per computation, we
  could get our answer 69 times faster with a
  quantum computer!

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The Quantum Computer (cont.)

• In order to ask many questions at once, we need a
  way to represent data at a quantum level.
• Just as with classical computers, we can use 0s
  and 1s the represent numbers in binary.
• One way to do this is via the spin of a
  fundamental particle (such as an electron).
• Many fundamental particles possess an inherent
  spin - they spin either west (clockwise) or
  east (counterclockwise).
• Thus, we could identify westward spin with 0 and
  eastward spin with 1, so for example, seven
  particles with spins (in order) east, east,
  west, east, west, west would represent the binary
  number 110100 (decimal number 104).

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The Quantum Computer (cont.)

• With seven particles, we then would be able to
  represent any number between 0 and 127.
• Using a classic computer, we would have to enter
  each of the numbers (one at a time) as a string
  of seven spinning particles.
• For a quantum computer, we would enter all 128
  numbers at once as a superposition of the 128
  different states (one per number).
• A natural question to ask is How do we achieve
  this superposition?
• The key to achieving superposition is that until
  we observe a spinning particle, it could be
  spinning east or west, so it is in a
  superposition of the two states.

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The Quantum Computer (cont.)

• Suppose we observe a particle and it is spinning
  west.
• We can change its spin by adding a sufficient
  amount of energy to the particle.
• If we add less energy, then the particle may
  change spin or may stay the same.
• Using the idea of Schrodingers cat, put the west
  spinning particle in a box, close the lid, and
  add a little bit of energy to the particle.
• Until we open the box, we wont know the
  particles spin, so the particle has entered a
  superposition of the states east and west.
• By performing the same operation with seven
  particles, we will achieve a superposition of the
  128 possible states!

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The Quantum Computer (cont.)

• In a traditional computer, a 0 or a 1 is called a
  bit, which is short for binary digit.
• Since a quantum computer deals with a
  superposition of a 0 and a 1, we call such an
  object a qubit, which is short for quantum bit.

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Applications of Quantum Computing (cont.)

• To give an even better idea of how powerful a
  quantum computer could be, suppose we were able
  compute with 250 qubits.
• Using the Fundamental Principal of Counting, the
  number of states in a superposition of these 250
  spinning particles would be 2250 which is about
  equal to 1.8 x 1075 different states (more than
  the number of particles in the universe)!
• Thus a quantum computer could perform over 1075
  simultaneous computations in a short amount of
  time!

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Applications of Quantum Computing (cont.)

• With this much computing power, one big
  application would be to factor large numbers
  quickly.
• In 1994, Peter Shor of ATT Bell Laboratories
  figured out how to program a quantum computer to
  factor a number larger than 129 digit number in a
  short amount of time (approximately 30 seconds).

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Applications of Quantum Computing (cont.)

• Here is why this is significant
• In 1977, Scientific American columnist Martin
  Gardner wrote an article entitled A New Kind of
  Cipher that Would Take Millions of Years to
  Break that announced the discovery of RSA
  cryptography to the world.
• In this article, he published a message encrypted
  with a 129 digit public key and offered a 100
  prize to the first person to decrypt the
  ciphertext.
• In 1994, 17 years later, a team of 600
  volunteers, using supercomputers and
  workstations, announced that they had found the
  factors of the public key and were able to
  decipher the message!

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Applications of Quantum Computing (cont.)

• Thus, a quantum computer can be used to very
  quickly crack RSA, which is what many people
  throughout the world rely on for transmitting
  messages securely!
• Another cryptography-related use of quantum
  computers is to search lists at high speed.
• Currently, another cryptographic scheme in use is
  DES, which relies on keys that, checking possible
  keys at a rate of one million per second, would
  take over 1000 years to crack.
• In 1996, Lov Grover, also at Bell Labs, found a
  way to program a quantum computer to find a DES
  key in about four minutes!

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Drawbacks to Quantum Computing

• So, if quantum computers are so great, what could
  possibly be wrong with them?
• One major issue is that we dont know how to make
  one!
• A lot of money has been invested into quantum
  computer research by government agencies, such as
  DARPA (Defense Advanced Research Projects
  Agency), but as Serge Heroche of the University
  of Paris IV put it (in 1998)
• Based on what we know about quantum computer
  technology, building one right now would be like
  carefully building the first layer of a house of
  cards and assuming that the next 15,000 layers
  are a mere formality!

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Dilberts Quantum Computer
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References

• The majority of this talk is based on material
  from Chapter 8 of The Code Book by Simon Singh,
  1999, Anchor Books.
• http//www.psd267.wednet.edu/kfranz/Science/Water
  Habitat/photojrnlmar00.htm
• http//micro.magnet.fsu.edu/optics/timeline/people
  /young.html
• http//www-groups.dcs.st-and.ac.uk/history/PictDi
  splay/Schrodinger.html
• http//en.wikipedia.org/wiki/Mirror,_Mirror_(Star_
  Trek)
• http//www.qubit.org/people/david/
• http//math.mit.edu/shor/
• http//www.csicop.org/si/9803/gardner.html