PLato said,"Look to the perfection of the heavens for truth," while Aristotle said "look around you at what is, if you would know the truth" To Remember: Eskesthai
Showing posts with label Quantum Computers. Show all posts
Showing posts with label Quantum Computers. Show all posts
In
his public lecture at Perimeter Institute on Oct. 5, 2015, Michele
Mosca (Institute for Quantum Computing, Perimeter Institute) explored
quantum technologies – those that already exist, and those yet to come –
and how they will affect our lives.
The modern double-slit experiment is a demonstration that light and matter can display characteristics of both classically defined waves and particles; moreover, it displays the fundamentally probabilistic nature of quantum mechanical phenomena. This experiment was performed originally by Thomas Young in 1801 (well before quantum mechanics) simply to demonstrate the wave theory of light and is sometimes referred to as Young's experiment.[1] The experiment belongs to a general class of "double path" experiments, in which a wave is split into two separate waves that later combine into a single wave. Changes in the path lengths of both waves result in a phase shift, creating an interference pattern. Another version is the Mach–Zehnder interferometer, which splits the beam with a mirror.Double-slit experiment
To some researchers, the experiments suggest that quantum objects are as definite as droplets, and that they too are guided by pilot waves — in this case, fluid-like undulations in space and time. These arguments have injected new life into a deterministic (as opposed to probabilistic) theory of the microscopic world first proposed, and rejected, at the birth of quantum mechanics. See: Have We Been Interpreting Quantum Mechanics Wrong This Whole Time?
In youtube example video given, I must say if you have ever seen
Taylor and Hulse's binary system, I couldn't help but see some
relation. Such rotation, would cause gravitational wave that seems to
hold the droplet in position for examination......but the gravitational
wave production, is an affect of this rotation so I am puzzled by this.
Natalie Wolchover is pretty good at her job, and I think drew attention
to the idea of a Bohemian mechanics/Pilot wave theory. This, as an
alteration of choice of quantum mechanics it became clear, how
interpretation was pervasive at the time between these two groups, as a
point of view. Not saying this is the case, but as I read I see the
division between the scientists as to how an interpretation arose
between them, some choose one way and others, another. And still they
did not discard the world of the two groups but leaned specifically to
one side over another.
As
de Broglie explained that day to Bohr, Albert Einstein, Erwin
Schrödinger, Werner Heisenberg and two dozen other celebrated
physicists, pilot-wave theory made all the same predictions as the
probabilistic formulation of quantum mechanics (which wouldn’t be
referred to as the “Copenhagen” interpretation until the 1950s), but
without the ghostliness or mysterious collapse. -Have We Been Interpreting Quantum Mechanics Wrong This Whole Time?
I am looking at the experiment itself as illustrated in my link to
youtube video of respective scientists given the relation and analogy
used. This is to see the aspect of their relation to something current
in our understanding "as observation," and something much more to it as
particle and wave together. Still trying to understand the analogy. In
the experiment, what leads the way, the wave, or the particle/droplet?
The "wave function" guides the particle/droplet, yes? Why of course, it
is called pilot-wave theory.
Before the experiment begins then, you know the particles state "as a
wave function," and given that this is already known, "the particle"
rides the wave function, is exemplary of the nature of the perspective
in the first place, as to what is already known. Hmmmm....sounds a
little confusing to me as I was seeing the waves in the experiment, but
given that such state of coalesce exists when experiment is done,
raises questions for me about the shaker as a necessity?
So cosmological you are looking to the past? You look up at the night
sky and when were all these messages received in the classical sense but
to be an observer of what happened a long time ago. You recognize the pathway as a wave function already before the
experimenter of the double slit even begins. It has a trajectory path
already given as the wave function is known with regard to A to B. These
are not probabilities then, if recognized as potential of the wave
function as already defining a pathway.
The pathway expressed as the pattern, had to already been established
as a causative event in the evolution in the recognition of a collision
course regarding any synchronized event located in the quantum world,
as a wave function pattern. You are dealing with a Bohemian
interpretation here.
***
On the flip side, I see spintronics, as a wave function giving
consideration to the y direction. It is a analogy that comes to mind
when I think of the fluid. Whether right or not, I see an association.
The idea, as a wave function is seen in regard to this chain as an illustration of the complexity of the fluid surface https://youtu.be/pWQ3r-2Xjeo
To go further then,
Known
as a major facet in the study of quantum hydrodynamics and macroscopic
quantum phenomena, the superfluidity effect was discovered by Pyotr
Kapitsa[1] and John F. Allen, and Don Misener[2] in 1937. It has since
been described through phenomenological and microscopic theories. The
formation of the superfluid is known to be related to the formation of a
Bose–Einstein condensate. This is made obvious by the fact that
superfluidity occurs in liquid helium-4 at far higher temperatures than
it does in helium-3. Each atom of helium-4 is a boson particle, by virtue of its zero spin.
Bold and underline added for emphasis
A
Bose–Einstein condensate (BEC) is a state of matter of a dilute gas of
bosons cooled to temperatures very close to absolute zero (that is, very
near 0 K or −273.15 °C). Under such conditions, a large fraction of
bosons occupy the lowest quantum state, at which point macroscopic
quantum phenomena become apparent.
So fast forward to the idealistic perception of the analog by
comparison in today's use against a backdrop of the theories and what do
we see?
Nevertheless,
they have proven useful in exploring a wide range of questions in
fundamental physics, and the years since the initial discoveries by the
JILA and MIT groups have seen an increase in experimental and
theoretical activity. Examples include experiments that have demonstrated interference between condensates due to wave–particle duality,[25]
the study of superfluidity and quantized vortices, the creation of
bright matter wave solitons from Bose condensates confined to one
dimension, and the slowing of light pulses to very low speeds using
electromagnetically induced transparency.[26] Vortices in Bose–Einstein
condensates are also currently the subject of analogue gravity research,
studying the possibility of modeling black holes and their related
phenomena in such environments in the laboratory. Experimenters have
also realized "optical lattices", where the interference pattern from
overlapping lasers provides a periodic potential. These have been used
to explore the transition between a superfluid and a Mott insulator,[27]
and may be useful in studying Bose–Einstein condensation in fewer than
three dimensions, for example the Tonks–Girardeau gas. -https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate#Current_research
-The theory of reinforcement learning provides a normative account1, deeply rooted in psychological2 and neuroscientific3 perspectives on animal behaviour, of how agents may optimize their control of an environment. To use reinforcement learning successfully in situations approaching real-world complexity, however, agents are confronted with a difficult task: they must derive efficient representations of the environment from high-dimensional sensory inputs, and use these to generalize past experience to new situations. Remarkably, humans and other animals seem to solve this problem through a harmonious combination of reinforcement learning and hierarchical sensory processing systems4, 5, the former evidenced by a wealth of neural data revealing notable parallels between the phasic signals emitted by dopaminergic neurons and temporal difference reinforcement learning algorithms3. While reinforcement learning agents have achieved some successes in a variety of domains6, 7, 8, their applicability has previously been limited to domains in which useful features can be handcrafted, or to domains with fully observed, low-dimensional state spaces. Here we use recent advances in training deep neural networks9, 10, 11 to develop a novel artificial agent, termed a deep Q-network, that can learn successful policies directly from high-dimensional sensory inputs using end-to-end reinforcement learning. We tested this agent on the challenging domain of classic Atari 2600 games12. We demonstrate that the deep Q-network agent, receiving only the pixels and the game score as inputs, was able to surpass the performance of all previous algorithms and achieve a level comparable to that of a professional human games tester across a set of 49 games, using the same algorithm, network architecture and hyperparameters. This work bridges the divide between high-dimensional sensory inputs and actions, resulting in the first artificial agent that is capable of learning to excel at a diverse array of challenging tasks.
***
This demo follows the description of the Deep Q Learning algorithm described in
Playing Atari with Deep Reinforcement Learning,
a paper from NIPS 2013 Deep Learning Workshop from DeepMind. The paper is a nice demo of a fairly
standard (model-free) Reinforcement Learning algorithm (Q Learning) learning to play Atari games.
In this demo, instead of Atari games, we'll start out with something more simple:
a 2D agent that has 9 eyes pointing in different angles ahead and every eye senses 3 values
along its direction (up to a certain maximum visibility distance): distance to a wall, distance to
a green thing, or distance to a red thing. The agent navigates by using one of 5 actions that turn
it different angles. The red things are apples and the agent gets reward for eating them. The green
things are poison and the agent gets negative reward for eating them. The training takes a few tens
of minutes with current parameter settings.
Over time, the agent learns to avoid states that lead to states with low rewards, and picks actions
that lead to better states instead.
***
Code for Human-Level Control through Deep Reinforcement Learning
The dispute between rationalism and empiricism concerns the extent to
which we are dependent upon sense experience in our effort to gain
knowledge. Rationalists claim that there are significant ways in which
our concepts and knowledge are gained independently of sense
experience. Empiricists claim that sense experience is the ultimate
source of all our concepts and knowledge.
Rationalists generally develop their view in two ways. First, they
argue that there are cases where the content of our concepts or
knowledge outstrips the information that sense experience can provide.
Second, they construct accounts of how reason in some form or other
provides that additional information about the world. Empiricists
present complementary lines of thought. First, they develop accounts
of how experience provides the information that rationalists cite,
insofar as we have it in the first place. (Empiricists will at times
opt for skepticism as an alternative to rationalism: if experience
cannot provide the concepts or knowledge the rationalists cite, then
we don't have them.) Second, empiricists attack the rationalists'
accounts of how reason is a source of concepts or knowledge. SEE: Markie, Peter, "Rationalism vs. Empiricism http://plato.stanford.edu/entries/rationalism-empiricism/", The Stanford Encyclopedia of Philosophy (Summer 2013 Edition), Edward N. Zalta (ed.),
Long before I had come to understand this nature of rationalism there
were already signs that such a journey was already being awakened. This was an
understanding for me as to the nature of what could be gained from the ability to visualize beyond empirical nature of our
journey into the sensible realm.
I guess in a such an awakening, as to what we know, there is the realization that what comes after helps to make that sense. So in a way one might like to see how rationalism together with Empiricism actually works. It is not in the sense that I might define one group of historical thinkers to contrast each other to say that one should excel over another, but to define how such a rationally sound person moves toward empiricism to understand the reality we created by experimentation and repeatability that empiricism enshrouds.
So this awakening while slow to materialize, comes from understanding something about the logic of the world and the definitions and architecture of that logical approach. To me in this day and age it has lead to some theory about which computational view could hold the idea about how we might see this reality. I am reticence to view this as a form of that reality. It is for what holds me back is a self evident moment using deducted features of our reasoning, which could move us to that moment of clarity.
The Empiricism Thesis: We have no source of knowledge in S or for the concepts we use in S other than sense experienceEmpircism -
The basis of this association(Rationalist, or, a Empiricist) is whether
one gains by a deductive method, or, an inductive method. A sense experience tells us, science as we know it, is inductive. We
must garner repeatable experiments to verify reality, a rationalist, by
logic and reason of theory alone. Verification, comes afterward. This for a rationalist is a deductive
something which can be true, can be "innate" before we accept the
inductive method means, that is it can be rationally ascertained. It is only after ward that such a process could be said to be true or false.
If the late character of our sources may incite us to doubt the
authenticity of this tradition, there remains that, in its spirit, it is
in no way out of character, as can be seen by reading or rereading what
Plato says about the sciences fit for the formation of philosophers in
book VII of the Republic, and especially about geometry at Republic,
VII, 526c8-527c11. We should only keep in mind that, for Plato,
geometry, as well as all other mathematical sciences, is not an end in
itself, but only a prerequisite meant to test and develop the power of
abstraction in the student, that is, his ability to go beyond the level
of sensible experience which keeps us within the "visible" realm, that
of the material world, all the way to the pure intelligible. And
geometry, as can be seen through the experiment with the slave boy in
the Meno (Meno, 80d1-86d2), can also make us discover the existence of
truths (that of a theorem of geometry such as, in the case of the Meno,
the one about doubling a square) that may be said to be "transcendant"
in that they don't depend upon what we may think about them, but have to
be accepted by any reasonable being, which should lead us into
wondering whether such transcendant truths might not exist as well in
other areas, such as ethics and matters relating to men's ultimate
happiness, whether we may be able to "demonstrate" them or not.See: Frequently Asked Questions about Plato by Bernard SUZANNE
When you examine deeply the very nature of your journey, then, you
come to realize what is hidden underneath "experience." So while being
an empiricist, it is necessary to know that such a joining with the
rationalist correlates with the reasoned only after the mentioned
experience. These are "corollary experiences," which serve to identify
that which had been identified long before the sensible world had been
made known.
Paradoxically,
it was Einstein who reluctantly introduced the notion of spontaneous
events, which might after all be the root of Bellʼs theorem. The lesson
for the future could, however, be that we should build the notion of locality on the operationally clear 'no-signalling' condition—the impossibility of transferring information faster than light. After all, this is all that theory of relativity requires.
Empiricism then is to validate as a corollary that which had been
cognate(maybe poor choice of word here but instead should use
cognition). This does not mean you stop the process, but to extend
the visionary possibility of that which can be cognitive....peering into
the very nature of reality. Becomes the " we should build the notion of locality on the operationally clear 'no-signalling' condition."
Here the question of entanglement raises it's head to ask what is really being trasmitted as the corrallary of information, as a direct physical connection in a computational system. In a quantum gravity scheme what is exchanged as a spin 2 graviton we might examine in the corollary of this no signalling condition but as a direct understanding of gravitational signalling.?
Such an examination reveals the Innate process with which we may already know "some thing," is awakened by moving into the world of science. While we consider such computational reality in context of a ontological question, then, such a journey may be represented as the geometry of the being which reveals a deeper question about the make-up of that reality.
Affective Field Theory of Emotion regarding sensory development may aid in the journey for
understanding the place with which "the idea/form in expression
arises," and that which is reasoned, beyond the related abstractions of
"such a beginning," by becoming the ideal, in the empiricist world.
I cannot say for certain and I speculate. Bucky balls then bring to mind
this architectural structure? Let me give you an example of a recent
discovery. I have to wonder if Bucky was a Platonist at heart......with grand ideas? Perhaps you recognze some Platonist idea about perfection as if mathematically a Tegmarkan might have found some truth? Some absolute truth? Perhaps a Penrose truth (Quasicrystal and Information)?
Aperiodic tilings serve as mathematical models for quasicrystals,
physical solids that were discovered in 1982 by Dan Shechtman[3] who
subsequently won the Nobel prize in 2011.[4] However, the specific local
structure of these materials is still poorly understood .Aperiodic tilings -
While one starts with a single point of entry......the whole process
from another perspective is encapsulated. So you might work from the
hydrogen spectrum as a start with the assumption, that this process in
itself is enclosed.
240 E₈ polytope vertices using 5D orthographic_projection to 2D using
5-cube (Penteract) Petrie_polygon basis_vectors overlaid on electron
diffraction pattern of an Icosahedron Zn-Mg-Ho Quasicrystal. E8_(mathematics) and Quasicrystals
At the same time one might understand the complexity of the issue?
By now it is known theoretically that quantum angular momentum of any
kind has a discrete spectrum, which is sometimes imprecisely expressed
as "angular momentum is quantized".Stern–Gerlach experiment -
***
So possibly a Photon polarization principle inherent in
a quantum description of the wave and such a principle inherent in the
use of photosynthesis to describe a property not just of the capability
of using sun light, but of understanding this principle biologically in
human beings? I actually have a example of this use theoretically as a
product. Maybe Elon Musk might like to use it?
Photonic molecules are a synthetic form of matter in which photons bind together to form "molecules".
According to Mikhail Lukin, individual (massless) photons "interact
with each other so strongly that they act as though they have mass". The
effect is analogous to refraction. The light enters another medium,
transferring part of its energy to the medium. Inside the medium, it
exists as coupled light and matter, but it exits as light.[1]
While I would like to make it easy for you, I can only leave a title for
your examination. "The Nobel Prize in Physics 1914 Max von Laue." Yes, but if it is understood that some correlate process can be
understood from "a fundamental position," as to the architecture of
matter, what would this light have to say about the component
structuralism of the information we are missing?
The idea is not new. From a science fiction point of view, StarTrek had
these units that when you were hungry or wanted a drink you would have
this object materialize in a microwave type oven? Not the transporter.
So, you have this 3d printer accessing all information about the
structure and access to the building blocks of all matter in energy,
funneled through this replicator.
***
When Bucky was waving his arm between the earth and the moon.....did he
know about the three body problem, or how to look at the space between
these bodies in another way. If people think this is not real, then you
will have to tell those who use celestial mechanics that they are using
their satellite trajectories all wrong.
Ephemeralization, a term coined by R. Buckminster Fuller,
is the ability of technological advancement to do "more and more with
less and less until eventually you can do everything with nothing".[1]
Fuller's vision was that ephemeralization will result in
ever-increasing standards of living for an ever-growing population
despite finite resources.
Exactly. So it was not just "hand waving" Buckminister Fuller is
alluding too, but some actual understanding to "more is less?" One
applies the principle then? See? I am using your informational video to
explain.
ARTEMIS-P1 is the first spacecraft to navigate to and perform
stationkeeping operations around the Earth-Moon L1 and L2 Lagrangian
points. There are five Lagrangian points associated with the Earth-Moon
system. ARTEMIS - The First Earth-Moon Libration Orbiter -
To do more with less, it has to be understood that distance crossed
needs minimum usage of fuel to project the satellite over a great
distance. So they use "momentum" to swing satellites forward?
This is a list of various types of equilibrium, the condition of a system in which all competing influences are balanced. List of types of equilibrium -
Who are we? And what is our role in the universe? Information technology
is radically changing not only how we deal with the world and make
sense of it, or interact with each other, but also how we look at
ourselves and understand our own existence and responsibilities.
Philosophy Professor Floridi ( @Floridi ) will discuss such impact of
information technology on our lives and on our self-understanding; he
will take us along the Copernican revolution, the Darwinian revolution,
the Freudian revolution right up to speed with the Turing revolution: a
world of inforgs in a global environment ultimately made of information.
Floridi will talk about expanding our ecological and ethical approach
to both natural and man-made realities, in order to cope successfully
with the new moral challenges posed by information technology. Ready for
some philosophy? You bet!
Princeton University physicists built a powerful imaging device called a
scanning-tunneling microscope and used it to capture an image of an
elusive particle that behaves simultaneously like matter and antimatter.
To avoid vibration, the microscope is cooled close to absolute zero and
is suspended like a floating island in the floor above. The setup
includes a 40-ton block of concrete, which is visible above the
researchers. The research team includes, from left, graduate student
Ilya Drozdov, postdoctoral researcher Sangjun Jeon, and professors of
physics B. Andrei Bernevig and Ali Yazdani. (Photo by Denise Applewhite, Office of Communications)
Majorana fermions are predicted to
localize at the edge of a topological superconductor, a state of matter
that can form when
a ferromagnetic system is placed in proximity to
a conventional superconductor with strong spin-orbit interaction. With
the
goal of realizing a one-dimensional topological
superconductor, we have fabricated ferromagnetic iron (Fe) atomic chains
on
the surface of superconducting lead (Pb). Using
high-resolution spectroscopic imaging techniques, we show that the onset
of
superconductivity, which gaps the electronic
density of states in the bulk of the Fe chains, is accompanied by the
appearance
of zero energy end states. This spatially
resolved signature provides strong evidence, corroborated by other
observations,
for the formation of a topological phase and
edge-bound Majorana fermions in our atomic chains. Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor
The unique capability of quantum mechanics to evolve alternative
possibilities in parallel is appealing and over the years a number of quantum
algorithms have been developed offering great computational benefits. Systems
coupled to the environment lose quantum coherence quickly and realization of
schemes based on unitarity might be impossible. Recent discovery of room
temperature quantum coherence in light harvesting complexes opens up new
possibilities to borrow concepts from biology to use quantum effects for
computational purposes. While it has been conjectured that light harvesting
complexes such as the Fenna-Matthews-Olson (FMO) complex in the green sulfur
bacteria performs an efficient quantum search similar to the quantum Grover's
algorithm the analogy has yet to be established. See: Evolutionary Design in Biological Quantum Computing
Spintronics (a neologism for "spin-based electronics"), also known as magnetoelectronics, is an emerging technology which exploits the quantum spin states of electrons as well as making use of their charge state. The electron spin itself is manifested as a two state magnetic energy system.
The discovery of giant magnetoresistance in 1988 by Albert Fert et al. and Peter Grünberg et al. independently is considered as the birth of spintronics.
David Awschalom explains how the spin of the electron could be exploited in completely new types of electronic circuits
If you are new to spintronics - or if you are wondering what all the excitement is about -- David Awschalom of the University of California, Santa Barbara provides a fantastic introduction to the field and explains how electron spin could be harnessed to create even denser computer memories and even quantum computers.
In this video interview Awschalom also outlines the challenges that must be overcome before we see the next generation of spintronics devices and explains how he is addressing some of these in his lab.
An artist's conception of the Majorana - a previously elusive subatomic
particle whose existence has never been confirmed - until now.
Dutch nano-scientists at the technological universities of
Delft and Eindhoven, say they have found evidence of the particle.
To find it, they devised miniscule circuitry around a
microscopic wire in contact with a semiconductor and a superconductor.
Lead researcher Leo Kouwenhoven.
SOUNDBITE (English), NANOSCIENTIST OF DELFT UNIVERSITY,
LEO KOUWENHOVEN, SAYING:
"The samples that we use for measuring the Majorana
fermions are really very small, you can see the holder of the sample,
the sample is actually inside here and if you zoom in, you can actually
see little wires and if you zoom in more, you see a very small
nano-meter scale sample, where we detected one pair of Majoranas."
When a magnetic field was applied along the the
'nanowire', electrons gathered together in synchrony as a Majorana
particle.
These subatomic particles could be used to encode
information, turning them into data to be used inside a tiny, quantum
computer.
SOUNDBITE (English), NANOSCIENTIST OF DELFT UNIVERSITY,
LEO KOUWENHOVEN, SAYING:
"The goal is actually to develop those nano-scale devices
into little circuits and actually make something like a quantum computer
out of it, so they have special properties that could be very useful
for computation, a particural kind of computation which we call quantum
computation, which would replace actually our current computers by
computers that are much more efficient than what we have now."
The Majorana fermion's existence was first predicted 75
years ago by Italian Ettore Majorana.
Probing the Majorana's particles could allow scientists to
understand better the mysterious realm of quantum mechanics.
Other groups working in solid state physics are thought to
be close to making similar announcements....heralding a new era in
super-powerful computer technology.
Were he alive today Majorana may well be amazed at the
sophisticated computer technology available to ordinary people in every
day life. But compared to the revolution his particle may be about to
spark, it will seem old fashioned in the not too distant future.
Jim Drury, Reuters
A Majorana fermion is a fermion that is its own anti-particle. The term is sometimes used in opposition to Dirac fermion, which describes particles that differ from their antiparticles. It is common that bosons (such as the photon) are their own anti-particle. It is also quite common that fermions can be their own anti-particle, such as the fermionic quasiparticles in spin-singlet superconductors (where the quasiparticles/Majorana-fermions carry spin-1/2) and in superconductors with spin-orbital coupling, such as Ir, (where the quasiparticles/Majorana-fermions do not carry well defined spins).
The concept goes back to Ettore Majorana's 1937 suggestion[1] that neutral spin-1/2 particles can be described by a real wave equation (the Majorana equation), and would therefore be identical to their antiparticle (since the wave function of particle and antiparticle are related by complex conjugation).
The difference between Majorana fermions and Dirac fermions can be expressed mathematically in terms of the creation and annihilation operators of second quantization. The creation operator γ†j creates a fermion in quantum state j, while the annihilation operator γj annihilates it (or, equivalently, creates the corresponding antiparticle). For a Dirac fermion the operators γ†j and γj are distinct, while for a Majorana fermion they are identical.
Elementary particle
No elementary particle is known to be a Majorana fermion. However, the nature of the neutrino is not yet definitely settled; it might be a Majorana fermion or it might be a Dirac fermion. If it is a Majorana fermion, then neutrinoless double beta decay is possible; experiments are underway to search for this type of decay.
The hypothetical neutralino of supersymmetric models is a Majorana fermion.
Quasiparticle
In superconducting materials, Majorana fermions can emerge as (non-fundamental) quasiparticles.[2](People
also name protected zero-energy mode as Majorana fermion. The
discussions in the rest of this article are actually about such
protected zero-energy mode, which is quite different from the
propagating particle introduced by Majorana.) The superconductor imposes electron hole symmetry on the quasiparticle excitations, relating the creation operator γ(E) at energy E to the annihilation operator γ†(−E) at energy −E. At the Fermi levelE=0, one has γ=γ†
so the excitation is a Majorana fermion. Since the Fermi level is in
the middle of the superconducting gap, these are midgap states. A quantum vortex in certain superconductors or superfluids can trap midgap states, so this is one source of Majorana fermions.[3][4][5]Shockley states at the end points of superconducting wires or line defects are an alternative, purely electrical, source.[6] An altogether different source uses the fractional quantum Hall effect as a substitute for the superconductor.[7]
It was predicted that Majorana fermions in superconductors could be used as a building block for a (non-universal) topological quantum computer, in view of their non-Abelian anyonic statistics.[8]
Experiments in superconductivity
In 2008 Fu and Kane provided a groundbreaking development by
theoretically predicting that Majorana fermions can appear at the
interface between topological insulators and superconductors.[9][10]
Many proposals of a similar spirit soon followed. An intense search to
provide experimental evidence of Majorana fermions in superconductors[11][12] first produced some positive results in 2012.[13][14] A team from the Kavli Institute of Nanoscience at Delft University of Technology in the Netherlands reported an experiment involving indium antimonide
nanowires connected to a circuit with a gold contact at one end and a
slice of superconductor at the other. When exposed to a moderately
strong magnetic field the apparatus showed a peak electrical conductance
at zero voltage that is consistent with the formation of a pair of
Majorana quasiparticles, one at either end of the region of the nanowire
in contact with the superconductor.[15]
This experiment from Delft marks a possible verification of independent theoretical proposals from two groups[16][17] predicting the solid state manifestation of Majorana fermions in semiconducting wires.
It is important to note that the solid state manifestations of
Majorana fermions are emergent low-energy localized modes of the system
(quasiparticles) which are not fundamental new elementary particles as
originally envisioned by Majorana (or as the neutrino would be if it
turns out to be a Majorana fermion), but are effective linear
combinations of half-electrons and half-holes which are topological
anyonic objects obeying non-Abelian statistics.[8] The terminology "Majorana fermion" is thus not a good nomenclature for these solid state Majorana modes.
References
^E. Majorana (1937). "Teoria simmetrica dell’elettrone e del positrone" (in Italian). Nuovo Cimento14: 171. English translation.
^ abC. Nayak, S. Simon, A. Stern, M. Freedman, and S. Das Sarma (2008). "Non-Abelian anyons and topological quantum computation". Reviews of Modern Physics80: 1083.
The Majorana experiment will search for neutrinoless double-beta decay of 76Ge. The discovery of this process would imply that the neutrino is a Majorana fermion (its own anti-particle) and allow a measurement of the effective Majorana neutrino mass.
The first stage of the experiment, the Majorana Demonstrator, will consist of 60kg of germanium crystal detectors in three cryostats. Half of these will be made from natural germanium and half from germanium enriched in 76Ge. The goals of the Demonstrator are to test a claim for measurement of neutrinoless double beta-decay by Klapdor-Kleingrothaus et al. (2006), to demonstrate a low enough background to justify the construction of a larger tonne-scale experiment, and to demonstrate the scalability of the technology to the tonne scale. The experiment will be located at the 4850 ft level of the Sanford Laboratory in Lead, South Dakota. See:
The Majorana neutrinoless double beta-decay experiment
The terms are not quite synonymous: "super-Turing computation"
usually implies that the proposed model is supposed to be physically
realizable, while "hypercomputation" does not.
Technical arguments against the physical realizability of hypercomputations have been presented.
A computational model going beyond Turing machines was introduced by Alan Turing in his 1938 PhD dissertation Systems of Logic Based on Ordinals.[2] This paper investigated mathematical systems in which an oracle was available, which could compute a single arbitrary (non-recursive) function from naturals to naturals. He used this device to prove that even in those more powerful systems, undecidability is still present. Turing's oracle machines are strictly mathematical abstractions, and are not physically realizable.[3]
Hypercomputation and the Church–Turing thesis
The Church–Turing thesis
states that any function that is algorithmically computable can be
computed by a Turing machine. Hypercomputers compute functions that a
Turing machine cannot, hence, not computable in the Church-Turing sense.
An example of a problem a Turing machine cannot solve is the halting problem.
A Turing machine cannot decide if an arbitrary program halts or runs
forever. Some proposed hypercomputers can simulate the program for an
infinite number of steps and tell the user whether or not the program
halted.
Hypercomputer proposals
A Turing machine that can complete infinitely many steps.
Simply being able to run for an unbounded number of steps does not
suffice. One mathematical model is the Zeno machine (inspired by Zeno's paradox).
The Zeno machine performs its first computation step in (say) 1 minute,
the second step in ½ minute, the third step in ¼ minute, etc. By
summing 1+½+¼+... (a geometric series) we see that the machine performs infinitely many steps in a total of 2 minutes. However, some[who?] people claim that, following the reasoning from Zeno's paradox, Zeno machines are not just physically impossible, but logically impossible.[4]
Turing's original oracle machines, defined by Turing in 1939.
In mid 1960s, E Mark Gold and Hilary Putnam independently proposed models of inductive inference (the "limiting recursive functionals"[5] and "trial-and-error predicates",[6] respectively). These models enable some nonrecursive sets of numbers or languages (including all recursively enumerable
sets of languages) to be "learned in the limit"; whereas, by
definition, only recursive sets of numbers or languages could be
identified by a Turing machine. While the machine will stabilize to the
correct answer on any learnable set in some finite time, it can only
identify it as correct if it is recursive; otherwise, the correctness is
established only by running the machine forever and noting that it
never revises its answer. Putnam identified this new interpretation as
the class of "empirical" predicates, stating: "if we always 'posit' that
the most recently generated answer is correct, we will make a finite
number of mistakes, but we will eventually get the correct answer.
(Note, however, that even if we have gotten to the correct answer (the
end of the finite sequence) we are never sure that we have the correct answer.)"[6]L. K. Schubert's 1974 paper "Iterated Limiting Recursion and the Program Minimization Problem" [7] studied the effects of iterating the limiting procedure; this allows any arithmetic
predicate to be computed. Schubert wrote, "Intuitively, iterated
limiting identification might be regarded as higher-order inductive
inference performed collectively by an ever-growing community of lower
order inductive inference machines."
A real computer (a sort of idealized analog computer) can perform hypercomputation[8] if physics admits general real variables (not just computable reals),
and these are in some way "harnessable" for computation. This might
require quite bizarre laws of physics (for example, a measurable physical constant with an oracular value, such as Chaitin's constant), and would at minimum require the ability to measure a real-valued physical value to arbitrary precision despite thermal noise and quantum effects.
A proposed technique known as fair nondeterminism or unbounded nondeterminism may allow the computation of noncomputable functions.[9]
There is dispute in the literature over whether this technique is
coherent, and whether it actually allows noncomputable functions to be
"computed".
It seems natural that the possibility of time travel (existence of closed timelike curves
(CTCs)) makes hypercomputation possible by itself. However, this is not
so since a CTC does not provide (by itself) the unbounded amount of
storage that an infinite computation would require. Nevertheless, there
are spacetimes in which the CTC region can be used for relativistic
hypercomputation.[10] Access to a CTC may allow the rapid solution to PSPACE-complete problems, a complexity class which while Turing-decidable is generally considered computationally intractable.[11][12]
In 1994, Hava Siegelmann
proved that her new (1991) computational model, the Artificial
Recurrent Neural Network (ARNN), could perform hypercomputation (using
infinite precision real weights for the synapses). It is based on
evolving an artificial neural network through a discrete, infinite
succession of states.[17]
The infinite time Turing machine is a generalization of the
Zeno machine, that can perform infinitely long computations whose steps
are enumerated by potentially transfinite ordinal numbers.
It models an otherwise-ordinary Turing machine for which non-halting
computations are completed by entering a special state reserved for
reaching a limit ordinal and to which the results of the preceding infinite computation are available.[18]
Jan van Leeuwen and Jiří Wiedermann wrote a 2000 paper[19] suggesting that the Internet should be modeled as a nonuniform computing system equipped with an advice function representing the ability of computers to be upgraded.
A symbol sequence is computable in the limit if there is a finite, possibly non-halting program on a universal Turing machine that incrementally outputs every symbol of the sequence. This includes the dyadic expansion of π and of every other computable real,
but still excludes all noncomputable reals. Traditional Turing machines
cannot edit their previous outputs; generalized Turing machines, as
defined by Jürgen Schmidhuber,
can. He defines the constructively describable symbol sequences as
those that have a finite, non-halting program running on a generalized
Turing machine, such that any output symbol eventually converges, that
is, it does not change any more after some finite initial time interval.
Due to limitations first exhibited by Kurt Gödel (1931), it may be impossible to predict the convergence time itself by a halting program, otherwise the halting problem could be solved. Schmidhuber ([20][21]) uses this approach to define the set of formally describable or constructively computable universes or constructive theories of everything. Generalized Turing machines can solve the halting problem by evaluating a Specker sequence.
In 1970, E.S. Santos defined a class of fuzzy logic-based "fuzzy algorithms" and "fuzzy Turing machines".[24]
Subsequently, L. Biacino and G. Gerla showed that such a definition
would allow the computation of nonrecursive languages; they suggested an
alternative set of definitions without this difficulty.[25] Jiří Wiedermann analyzed the capabilities of Santos' original proposal in 2004.[26]
Dmytro Taranovsky has proposed a finitistic
model of traditionally non-finitistic branches of analysis, built
around a Turing machine equipped with a rapidly increasing function as
its oracle. By this and more complicated models he was able to give an
interpretation of second-order arithmetic.[27]
Analysis of capabilities
Many hypercomputation proposals amount to alternative ways to read an oracle or advice function embedded into an otherwise classical machine. Others allow access to some higher level of the arithmetic hierarchy. For example, supertasking Turing machines, under the usual assumptions, would be able to compute any predicate in the truth-table degree containing or . Limiting-recursion, by contrast, can compute any predicate or function in the corresponding Turing degree, which is known to be . Gold further showed that limiting partial recursion would allow the computation of precisely the predicates.
neural networks based on real numbers (Hava Siegelmann)
Criticism
Martin Davis, in his writings on hypercomputation [39][40]
refers to this subject as "a myth" and offers counter-arguments to the
physical realizability of hypercomputation. As for its theory, he argues
against the claims that this is a new field founded in 1990s. This
point of view relies on the history of computability theory (degrees of
unsolvability, computability over functions, real numbers and ordinals),
as also mentioned above. Andrew Hodges wrote a critical commentary[41] on Copeland and Proudfoot's article[1].
^Alan Turing, 1939, Systems of Logic Based on Ordinals Proceedings London Mathematical Society Volumes 2–45, Issue 1, pp. 161–228.[1]
^"Let
us suppose that we are supplied with some unspecified means of solving
number-theoretic problems; a kind of oracle as it were. We shall not go
any further into the nature of this oracle apart from saying that it
cannot be a machine" (Undecidable p. 167, a reprint of Turing's paper Systems of Logic Based On Ordinals)
^ abcHilary Putnam (1965). "Trial and Error Predicates and the Solution to a Problem of Mostowksi". Journal of Symbolic Logic30 (1): 49–57. doi:10.2307/2270581. JSTOR2270581.
^Arnold Schönhage, "On the power of random access machines", in Proc. Intl. Colloquium on Automata, Languages, and Programming (ICALP), pages 520-529, 1979. Source of citation: Scott Aaronson, "NP-complete Problems and Physical Reality"[2] p. 12
^Edith Spaan, Leen Torenvliet and Peter van Emde Boas (1989). "Nondeterminism, Fairness and a Fundamental Analogy". EATCS bulletin37: 186–193.
^Hajnal Andréka, István Németi and Gergely Székely, Closed Timelike Curves in Relativistic Computation, 2011.[3]
^Todd A. Brun, Computers with closed timelike curves can solve hard problems, Found.Phys.Lett. 16 (2003) 245-253.[4]
^S. Aaronson and J. Watrous. Closed Timelike Curves Make Quantum and Classical Computing Equivalent [5]
^Hogarth,
M., 1992, ‘Does General Relativity Allow an Observer to View an
Eternity in a Finite Time?’, Foundations of Physics Letters, 5, 173–181.
^István Neméti; Hajnal Andréka (2006). "Can General Relativistic Computers Break the Turing Barrier?". Logical
Approaches to Computational Barriers, Second Conference on
Computability in Europe, CiE 2006, Swansea, UK, June 30-July 5, 2006.
Proceedings. Lecture Notes in Computer Science. 3988. Springer. doi:10.1007/11780342.
^ abJan van Leeuwen; Jiří Wiedermann (September 2000). "On Algorithms and Interaction". MFCS '00: Proceedings of the 25th International Symposium on Mathematical Foundations of Computer Science. Springer-Verlag.
^Jürgen Schmidhuber (2000). "Algorithmic Theories of Everything". Sections
in: Hierarchies of generalized Kolmogorov complexities and
nonenumerable universal measures computable in the limit. International
Journal of Foundations of Computer Science ():587-612 (). Section 6 in:
the Speed Prior: A New Simplicity Measure Yielding Near-Optimal
Computable Predictions. in J. Kivinen and R. H. Sloan, editors,
Proceedings of the 15th Annual Conference on Computational Learning
Theory (COLT ), Sydney, Australia, Lecture Notes in Artificial
Intelligence, pages 216--228. Springer, .13 (4): 1–5. arXiv:quant-ph/0011122.
^Davis, Martin, Why there is no such discipline as hypercomputation, Applied Mathematics and Computation, Volume 178, Issue 1, 1 July 2006, Pages 4–7, Special Issue on Hypercomputation
^Davis, Martin (2004). "The Myth of Hypercomputation". Alan Turing: Life and Legacy of a Great Thinker. Springer.
L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and Real Computation, Springer-Verlag 1997. General development of complexity theory for abstract machines that compute on real numbers instead of bits.
Cooper, S. B.; Odifreddi, P. (2003). "Incomputability in Nature". In S. B. Cooper and S. S. Goncharov. Computability and Models: Perspectives East and West. Plenum Publishers, New York, Boston, Dordrecht, London, Moscow. pp. 137–160.
Copeland, J. (2002) Hypercomputation, Minds and machines, v. 12, pp. 461–502
or 
. Limiting-recursion, by contrast, can compute any predicate or function in the corresponding Turing degree, which is known to be 
. Gold further showed that limiting partial recursion would allow the computation of precisely the 
predicates.
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