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Hypercomputation. Computing beyond Turing machines. 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.”.
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Hypercomputation Computing beyond Turing machines
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.”
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?
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
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.
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.
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.
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?
Double Slit Experiment • A single particle traveling through two slits creates interference patterns with itself.
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.
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.
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?
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 wasn’t it Schrödinger's dog?)
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.
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.
Uncertainty Principle • Tunneling • Quantum Superposition
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.
In other words, Kieu claims to have discovered uncomputable problems that are actually computable with the help of quantum mechanics.
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.
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.
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.
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.
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.
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?
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.
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.
ref:W. Heitmann and G. Nimtz, Phys Lett A196, 154 (1994);A. Enders and G. Nimtz, Phys Rev E48, 632 (1993).
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.
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.
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.
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.
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.
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."
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."
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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)!!
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.
