The document discusses hypercomputation and the possibility of computing beyond Turing machines, questioning the Church-Turing thesis by exploring concepts like quantum mechanics, wave-particle duality, and Schrödinger's cat. It examines various theoretical devices and algorithms, including Kieu's claims of quantum algorithms that could solve problems deemed unsolvable by classical means, and suggests that quantum mechanics may allow for new forms of computation. However, while quantum phenomena have practical implications, the consensus remains that quantum computation does not lead to hypercomputation, with limits imposed by established physical laws.

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.  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
Possibilities:

5. But the world is a strange place
so it may be possible someday
 Wave-particle duality
 Double slit experiment
 Schrödinger's cat
 Kieu’s algorithm for Hilbert’s 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.  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 wasn’t it Schrödinger's dog?)
Schrödinger's cat

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.

16.  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 problem
Boris 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.  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.
Faster than light travel

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.  ref:
W. 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 Wang’s 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 AT&T'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.

51.  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?
Conclusion