Hypercomputation

1
Hypercomputation

• Computing beyond Turing machines

2
Hypercomputation

• According to the Church Turing thesis, anything
  that is computable is computable by a Turing
  machine.
• But how do we really know this? Maybe some
  devices are more powerful. This is the topic of
  hypercomputation.

3
Hypercomputation

• If some computing device were more powerful than
  a Turing machine, maybe we could solve the
  halting problem and other problems that are
  currently classified as unsolvable.
• But how might such a device work?

4
Possibilities

• Compute with infinite precision real numbers
• Quantum mechanical system using an infinite
  superposition of states (Kieu)
• A TM that keeps getting smaller and smaller,
  faster and faster
• None of these seem possible at present

5
But the world is a strange place so it may be
possible someday

• Wave-particle duality
• Double slit experiment
• Schrödinger's cat
• Kieus algorithm for Hilberts Tenth Problem
• Faster than light travel or information transfer?
  See this link.

6
Wave Particle Duality

• In quantum mechanics, the wave-particle duality
  is explained as follows every system and
  particle is described by wave functions which
  encode the probability distributions of all
  measurable variables. The position of the
  particle is one such variable. Before an
  observation is made the position of the particle
  is described in terms of probability waves which
  can interfere with each other.

7
Wave Particle Duality

• After measurement the position of the particle
  collapses to one location, the probability of
  each location determined by the wave probability
  function.

8

• In fact according to quantum mechanics the
  physical world is probabilistic and not
  deterministic
• The future is not completely determined by the
  past
• Leaves room for free will philosophically
• Differs from Newtonian mechanics which is
  deterministic
• Do electrons have free will?

9
Double Slit Experiment

• A single particle traveling through two slits
  creates interference patterns with itself.

10
Schrödinger's cat

• Schrödinger's cat is a thought experiment devised
  by Erwin Schrödinger that attempts to illustrate
  the incompleteness of the theory of quantum
  mechanics when going from subatomic to
  macroscopic systems.

11
Schrödinger's cat

• A cat is placed in a sealed box. Attached to the
  box is an apparatus containing a radioactive
  nucleus and a canister of poison gas. When the
  nucleus decays, it emits a particle that triggers
  the apparatus, which opens the canister and kills
  the cat.

12

• According to quantum mechanics, the nucleus is
  described as a superposition (mixture) of
  "decayed nucleus" and "undecayed nucleus".
  However, when the box is opened the experimenter
  sees only a "decayed nucleus/dead cat" or a
  "undecayed nucleus/living cat." The question is
  when does the system stop existing as a mixture
  of states and become one or the other?

13
Schrödinger's cat

• The purpose of the experiment is to illustrate
  that quantum mechanics is incomplete without some
  rules to describe when the wavefunction collapses
  and the cat becomes dead or alive instead of a
  mixture of both.
• (Why wasnt it Schrödinger's dog?)

14

• Curiously, all of this has some practical use in
  quantum cryptography. It is possible to send
  light that is in a superposition of states down a
  fiber optic cable. Placing a wiretap in the
  middle of the cable which intercepts and
  retransmits the transmission will collapse the
  wavefunction (in the Copenhagen interpretation,
  "perform an observation") and cause the light to
  fall into one state or another.

15

• By performing statistical tests on the light
  received at the other end of the cable, one can
  tell whether it remains in the superposition of
  states or has already been observed and
  retransmitted.

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• Uncertainty Principle
• Tunneling
• Quantum Superposition

17
Quantum Leaps

• "We dispute the Turing-Church thesis by showing
  that there exist computable functions --
  computable by executing well-defined quantum
  mechanical procedures in a finite manner -- that
  are not Turing-computable," Kieu claims in a
  recent paper on the topic.

18

• In other words, Kieu claims to have discovered
  uncomputable problems that are actually
  computable with the help of quantum mechanics.

19
Quantum Algorithm for Hilbert's Tenth Problem
(Kieu)

• We explore in the framework of Quantum
  Computation the notion of Computability, which
  holds a central position in Mathematics and
  Theoretical Computer Science.

20

• A quantum algorithm for Hilbert's tenth problem,
  which is equivalent to the Turing halting problem
  and is known to be mathematically noncomputable,
  is proposed where quantum continuous variables
  and quantum adiabatic evolution are employed.

21

• If this algorithm could be physically
  implemented, as much as it is valid in
  principle--that is, if certain hamiltonian and
  its ground state can be physically constructed
  according to the proposal--quantum computability
  would surpass classical computability as
  delimited by the Church-Turing thesis.

22

• It is thus argued that computability, and with it
  the limits of Mathematics, ought to be determined
  not solely by Mathematics itself but also by
  Physical Principles.

23
The quantum algorithm of Kieu does not solve the
Hilbert's tenth problemBoris Tsirelson

• Recently T. Kieu 1 claimed a quantum algorithm
  computing some functions beyond the Church-Turing
  class. He notes that "it is in fact widely
  believed that quantum computation cannot offer
  anything new about computability" and claims the
  opposite.

24

• However, his quantum algorithm does not work,
  which is the point of my short note. I still
  believe that quantum computation leads to new
  complexity but retains the old computability.
• Who is right, Kieu or Tsirelson?

25
Faster than light travel

• Yet a group of physicists have performed
  experiments which seem to suggest that FTL
  communication by quantum tunneling is possible.
  They claim to have transmitted Mozart's 40th
  Symphony through a barrier 11.4cm wide at a speed
  of 4.7c. Their interpretation is, of course, very
  controversial.

26

• Most physicists say this is a quantum effect
  where no information can actually be passed at
  FTL speeds because of the Heisenberg uncertainty
  principle. If the effect is real it is difficult
  to see why it should not be possible to transmit
  signals into the past by placing the apparatus in
  a fast moving frame of reference.

27

• refW. Heitmann and G. Nimtz, Phys Lett A196,
  154 (1994)A. Enders and G. Nimtz, Phys Rev E48,
  632 (1993).

28
Light Exceeds Its Own Speed Limit, or Does It?

• In the most striking of the new experiments by
  Lijun Wang of Princeton a pulse of light that
  enters a transparent chamber filled with
  specially prepared cesium gas is pushed to speeds
  of 300 times the normal speed of light. That is
  so fast that, under these peculiar circumstances,
  the main part of the pulse exits the far side of
  the chamber even before it enters at the near
  side.

29

• It is as if someone looking through a window from
  home were to see a man slip and fall on a patch
  of ice while crossing the street well before
  witnesses on the sidewalk saw the mishap occur--a
  preview of the future.

30

• Dr. Chiao, whose own research laid some of the
  groundwork for the experiment, added that
  "there's been a lot of controversy" over whether
  the finding means that actual information--like
  the news of an impending accident--could be sent
  faster than c, the velocity of light. But he said
  that he and most other physicists agreed that it
  could not.

31

• A paper on the second new experiment, by Daniela
  Mugnai, Anedio Ranfagni and Rocco Ruggeri of the
  Italian National Research Council, described what
  appeared to be slightly faster-than-c propagation
  of microwaves through ordinary air, and was
  published in the May 22 issue of Physical Review
  Letters.

32

• The overall result of Wangs experiment is an
  outgoing wave exactly the same in shape and
  intensity as the incoming wave the outgoing wave
  just leaves early, before the peak of the
  incoming wave even arrives.
• As most physicists interpret the experiment, it
  is a low-intensity precursor (sometimes called a
  tail, even when it comes first) of the incoming
  wave that clues the cesium chamber to the
  imminent arrival of a pulse.

33

• Someone who looked only at the beginning and end
  of the experiment would see only a pulse of light
  that somehow jumped forward in time by moving
  faster than c.
• "The effect is really quite dramatic," Dr.
  Steinberg said. "For a first demonstration, I
  think this is beautiful."

34

• But it really wouldn't allow anyone to send
  information faster than c, said Peter W. Milonni,
  a physicist at Los Alamos National Laboratory.
• "The information is already there in the leading
  edge of the pulse," Dr. Milonni said. "You can
  get the impression of sending information
  superluminally even though you're not sending
  information."

35

• Not all physicists agree that the question has
  been settled, though. "This problem is still
  open," said Dr. Ranfagni of the Italian group,
  which used an ingenious set of reflecting optics
  to create microwave pulses that seemed to travel
  as much as 25 faster than c over short
  distances.

36

• At least one physicist, Dr. Guenter Nimtz of the
  University of Cologne, holds the opinion that a
  number of experiments, including those of the
  Italian group, have in fact sent information
  superluminally. But not even Dr. Nimtz believes
  that this trick would allow one to reach back in
  time. He says, in essence, that the time it takes
  to read any incoming information would fritter
  away any temporal advantage, making it impossible
  to signal back and change events in the past.

37

• If we could send information into the past then
  we might solve the halting problem -- whenever
  the machine M halts, send a messsage back to a
  specified time t saying that it halts.
• If at time t no message is received then one
  knows M did not halt.

38

• Quantum entanglement. See this link and this
  link. Action at a distance.
• Quantum computation does not lead to
  hypercomputation. But it may lead to fast
  computation.

39
Quantum Entanglement

• At risk of oversimplification, QE is when the
  fate of two or more particles become bound
  together. A change in one entangled particle
  results in an INSTANT change in the other
  particle as well, no matter how far away it is -
  even at the opposite end of the universe.

40
Quantum Entanglement

• In the 1970s, physicist Alan Aspect successfully
  ran a version of the EPR experiment stretched
  across a space the size of a basketball court and
  showed that quantum entanglement in fact does
  exist. With this one experiment, the possibility
  of building a quantum computer seized the
  imagination of physicists.

41
Quantum Entanglement

• A computer based on quantum entanglement would
  have no limits at all on how fast it could
  perform logical switching operations since it
  would use "spooky action at a distance" instead
  of electrons or light.

42
Quantum Entanglement

• A team gathered two clouds of cesium gas, each
  containing about a trillion atoms, into separate,
  sealed vessels. They then shined a laser through
  both clouds. For a split second, the clouds
  became entangled, and magnetic changes in one
  instantly affected the other. The previous
  entanglement record was a mere four atoms.

43

• The development could lead to the creation of
  computers and communications networks that
  operate much faster than anything that's
  available today, says Peter Handel, a physics
  professor at the University of Missouri in St.
  Louis. "Information encoded in photons could be
  transmitted to places without sending them across
  space," he says.

44

• Quantum entanglement could also allow matter to
  be transported from one location to another by
  instantly duplicating the properties of one
  object in another place. Other researchers,
  however, are skeptical about quantum
  entanglement's sci-fi aspects. "You can't
  transfer information faster than the speed of
  light, that's an immutable law of physics," warns
  Randall Hulet, a physics and astronomy professor
  at Rice University in Houston.

45
Quantum Computation

• In a quantum computer, the fundamental unit of
  information (called a quantum bit or qubit), is
  not binary but rather more quaternary in nature.
  This qubit property arises as a direct
  consequence of its adherence to the laws of
  quantum mechanics which differ radically from the
  laws of classical physics.

46
Quantum Computation

• A qubit can exist not only in a state
  corresponding to the logical state 0 or 1 as in a
  classical bit, but also in states corresponding
  to a blend or superposition of these classical
  states. In other words, a qubit can exist as a
  zero, a one, or simultaneously as both 0 and 1,
  with a numerical coefficient representing the
  probability for each state.

47

• For example, a system of 500 qubits, which is
  impossible to simulate classically, represents a
  quantum superposition of as many as 2500 states.
  Any quantum operation on that system --a
  particular pulse of radio waves, for instance,
  whose action might be to execute a controlled-NOT
  operation on the 100th and 101st qubits-- would
  simultaneously operate on all 2500 states.

48

• Hence with one fell swoop, one tick of the
  computer clock, a quantum operation could compute
  not just on one machine state, as serial
  computers do, but on 2500 machine states at once!
  Eventually, however, observing the system would
  cause it to collapse into a single quantum state
  corresponding to a single answer, a single list
  of 500 1's and 0's, as dictated by the
  measurement axiom of quantum mechanics.

49

• The reason this is an exciting result is because
  this answer, derived from the massive quantum
  parallelism achieved through superposition, is
  the equivalent of performing the same operation
  on a classical super computer with 10150
  separate processors (which is of course
  impossible)!!

50

• Peter Shor, a research and computer scientist at
  ATT's Bell Laboratories in New Jersey, provided
  such an application of quantum computers by
  devising the first quantum computer algorithm.
  Shor's algorithm harnesses the power of quantum
  superposition to rapidly factor very large
  numbers (on the order 10200 digits and greater)
  in a matter of seconds.
• But large quantum computers have not yet been
  built and may be very hard to make.

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Conclusion

• With so much strangeness in the world, and much
  of it even having practical applications, who
  knows whether a computing device more powerful
  than Turing machines is possible?