Science-III: What is a Quantum Of Information?
INTRODUCTION
The identification of the quantum
of information has no satisfactory definition and appears to be a very difficult subject to pin down: almost like the Bigfoot or Sasquatch, the concept of quantum information quanta remain elusive.
In stark contrast, computational information theory, is defined using the bit, which is really just a nomenclature for a digit, whether a 0 or a 1, in the binary representation system. From this concept, analogs such as Qubit (aka the Quantum Bit) for representing two possible states and the Qutrit for three possible states as well as Qudit for multiple states have arisen in order to produce some language to talk about information. The basic assumption underlying Quantum Science is that energy somehow corresponds to information because it is energy that is needed in order to store information at the level: this energy, itself, must be at minimum, quantized according to the minimum energy level. So the model we have is implicitly tied to the physicality of information storage. The question here is whether this assumption is actually valid, useful and productive. Must information quanta be necessarily based on energy?
One basic problem starts when you look for a definition of information that everyone can agree on. You will find that there are a very large and diverse if not completely different definitions of information on the web, all competing to be the right definition in the literature.
In the Quantum context, the observer is a participant in the measurement and therefore information becomes subjective. In the alternative theories about the Quantum context, such as the Bohm Pilot Wave theory, then the observer is not a participant and information becomes objective.
So the first difficulty is picking whether information is subjective, objective, or indeterminate between subjective and objective reference (i.e. interactive). Is the Quantum context entangled or separate from its environment in terms of a systems view? According to Bohm, it is objective. According to Schrödinger it is subjective (and hence, Schrödinger's cat).
Let us take a trip back in time to Bernard Patten, who in 1980, wrote a wonderfully inspiring paper entitled "ENVIRONS: Relativistic Elementary Particles for the Environment" which you can find online here: Environs
Bernard Patten developed a model of information for studying ecosystems based on mathematical systems theory where concept an organism interacting with its environment formed a single
integral system constituting a single ecological unit (the ‘fundamental
particle’ of ecology) called the "environ". Later, he refined his concepts into six parts: holon, taxon, creaon, genon, environ, and enviroplasm.
These concepts were used to develop an iterative, interacting, systems model of ecology. We will, shortly, see how these items entwine in a prospective definition for a quantum of information.
But before we can do that, we need to take another side road: infons. The unit of information, an infon, is as close a word as you can get to the notion of a quantum of information.
Niels Bohr stated, for his complementarity theory, that "It is not possible to make one unambiguous picture (model) of reality, as uncertainty limits our knowledge". That this uncertainty is caused by the influence of the observer as participant is inevitable in Quantum systems and therefore, implies that the character of the nature of information is that it is forever incomplete and uncertain.
Stonier developed a model based on the idea of the infon as pure quantum of information and with a status as fundamental as that of the fermions and bosons. Devlin uses the term infon for his conception of information as well though the model proposed is different though equally valid.
The ontological status of current theories is that they use an implicit state representation and that the configuration of states is equivalent to the permutation of bits. Hence, states and bits become equivalent as structural system descriptions. The problems arise when trying to render kinetic systems descriptions because then one has to choose a discrete time and a transition method to say how one state transforms into another.
I have run out of time as I work to finalize some code and other
things ... oh well, [MORE NEXT TIME ON THE IDEA OF A QUANTUM OF
INFORMATION].
Sunday, October 26, 2014
Sunday, October 12, 2014
Science-II: Structures for Data Representations for Quantum Computing
Science-II: Structures for Data Representations for Quantum Computing and Classical Systems
Introduction
Entanglement is a terribly explained subject and much of it has to do with the plethora of analogies that try to explain it - I will not claim to do any better but I make an attempt to provide an alternative view.
Shrodinger coined the term in 1935.
You can find a really great introduction to the concept of entanglement at the Stanford link: Quantum Entanglement explained by the Stanford Encyclopedia of Philosophy
That is the best explanation available on the web and draws heavily from Shrodinger's original - you cannot do better. Mine is not better. But if that's too long, then, here's mine, a different take.
Let me start with an observation: all the explanations that you typically find when searching for an explanation of entanglement lead to some sort of statement about the existence of things that you can "see" (or observe or measure) and the concept of the "state" of quantum system, usually particles of some kind.
These explanations fail to address the fact that most folks think in terms of "sets" of objects and "observations" as some form of properties in the set. Nothing could be further away and more misleading when it comes to quantum systems since quantum systems are not related to sets of objects but are rather related to the possibilities for describing objects relative to you, their observer. These possibilities are infinite and are described in ways that take those infinities of possibilities into account - they are described by a choice of functions (and that universe of choices by which to choose functions that can describe the objects is itself infinite). Think of a description like this: there are an infinite number of ways to write a description for the number "7" - here are a few:
1. 0 + 7 = 7
2. 3 + 4 = 7
3. 1 + 2 + 4 = 7
4. 0 + 0 + 1 + 6 = 7
5. etc ...
But now I am diving into an area that I will defer so back to entanglement. Some popular descriptions, as I noted, lead you immediately into trouble because the counter-intuitive nature is difficult while thinking of a quantum system like a classical system (which is thought of like a set of things that you can describe with certitude). Quantum systems are spaces of inter-relating entities whose futures are created in part by their past interactions and your choice of how and what you wish to observe. Those choices lead to connected behaviours that Shrodinger called "entangled".
Here are some of the the other popular explanations, for example:
Science Daily Definition of Entanglement
Wikipedia's Definition of Quantum Entanglement
So I offer the following instead: a quantum system is not a set, but, a weird kind of "flow" or "flows". Think of air flow - you cannot see it but a flag waves in the wind. The quantum flows are invisible - in fact, they are so very invisible that even those qualified in the art cannot picture them so whatever I share with you is going to be a hack. Reading my description in the last few sentences, I am already in despair! Let me try again:
Another way to think of it is this: imagine two flags, waving in the wind. Certainly, they will not wave in opposite directions! Their waviness will in some ways appear to be similar, but, not identical. However, if one flag is waving this way, the other flag might be waving that way, and, there may appear to be a kind of regularity. In other words, because we have a the wind, blowing in the same direction, and generally with the same forces, both flags wave in a similar, correlated way. But this is not Quantum. This is Classical. Complicated classical but not quantum.
Quantum entanglement is not like that kind of correlation with the "flows" being a kind of wind of motions and the particles being like the flags. Quantum entanglement is more like the idea that I wanted to see a flag, so I took a bike ride and saw two flags blowing in the wind *because* I wanted to see a flag and took a bike ride --- yes, I know, this sounds weird.
This example or analogy, which it is not of entanglement, is difficult to picture because it is as if I am a participant in creating the outcome - well, actually, I am. That is a quantum feature (my being a participant in creating the outcome).
Here's another one: toss a coin. It can land heads or tails. If I toss the coin and it lands "heads" you know that the other side is "tails" that it has landed on. Imagine now two sheets of paper: one has the words "heads" on one and the other has the word "tails". If I put the papers into two separate envelopes and I ask you to go home with the sealed envelope, you don't know if it has heads or tails written on it. But, if I call you and tell you that my envelope has "heads" then you never need to open your envelope because you have the information and you can infer that the contents of the envelope is "tails". Put another way, imagine you have a coin, say a nickel, and I have a coin, say a quarter and we toss them: every single time, without fail, when your nickel lands on "heads", my quarter just so happens to land on "tails" and vice versa: so it is as if they are invisibly connected but not in any way we can see. If you go home and I go to my home and we repeat the coin tosses, the results are still the same - we can call each other to check on it, after we have tossed the coins and they have landed. Mine lands "heads" and yours is, of course, "tails".
Quantum entanglement is not like this either, but I am trying.
Let me try again: things that you can say get related because of an association or reason might be as disparate as the concept of carrying an umbrella out into sunny skies because the news reported it might rain - this correlation or association between two different things, one of which is observed, you carrying and umbrella, and the other which is hidden, the rain, because the skies are sunny, is like getting closer to the idea of entanglement.
But even this metaphor is not entanglement.
I guess you see where I am going with this. There is a kind of connection that is like a correlation or association but does not really involve anything other than that some information exists. How that information is related to the coin toss, or the umbrella and the rain, is not known - only that it is repeatable and regular in how we can, as outsiders, see the outcomes.
The principle that is strange in entanglement is that of "nonlocality". Imagine our two coins sent with us in opposite directions and the state of one of them is altered (like I turn my quarter to "heads"), the second instantly alters its state (your nickel becomes "tails") in response no matter how far apart the two may be. This physical example is just not realistic but that is what happens at the microscopic level (e.g. photons that are entangled but going in opposite directions).
Albert Einstein he called it "spooky action at a distance" because there was no apparent physical connection underlying the phenomena and no apparent way to explain it.
Some entanglement phenomena are only just emerging such as as association between time and energy: "Now, physicists are unmasking a more fundamental source for the arrow of time: Energy disperses and objects equilibrate, they say, because of the way elementary particles become intertwined when they interact — a strange effect called “quantum entanglement.” ... “Finally, we can understand why a cup of coffee equilibrates in a room,” said Tony Short, a quantum physicist at Bristol. “Entanglement builds up between the state of the coffee cup and the state of the room.” - quoted from Time’s Arrow Traced to Quantum Source
So now we come to the really, really tough part of this blog: how can we represent entanglement?
1. Conceptual Entanglement - we can look at concepts of entanglement for computer systems by looking at the semantics of concept combinations and their representations. A good place to start is Quantum Entanglement in Concept Combinations
2. Representating Entanglement as an artifact in a graph structure: a paper to ideate with is: Entangled graphs: Bipartite entanglement in multi-qubit systems
3. Entanglement polytopes: there are geometries that may prove useful as representational structures and the following two papers should prove sufficient to get you started: Convex polytopes and quantum separability and, the paper here: Entanglement Polytopes
4. Hidden Symmetries, Hidden Variable, Knot Topological and representations of entanglements that are not really "quantum" but are quantum inspired or quasi-quantum (since the work of several physicists has shown that there are no hidden variables): Quantum logic as superbraids of entangled qubit world lines ; Experimental Demonstration of the Topological Phase for Entangled Qubits ; and other references that are simply just way out on the edge (and perhaps a new mathematical model, or physics will be needed to produce a better representation concept for entanglement); and finally (at least on this short list):
5. Using Alternative Probabilistic models such as Quantum Bayesianism to specify models in which entanglement can be represented.
Well, that's all for now, until next time, when we look at Science of Quanta. What is a quantum of information? Is there a quantum of knowledge? Of wisdom? And how could we represent such things, if it is even possible?
Introduction
Entanglement is a terribly explained subject and much of it has to do with the plethora of analogies that try to explain it - I will not claim to do any better but I make an attempt to provide an alternative view.
Shrodinger coined the term in 1935.
You can find a really great introduction to the concept of entanglement at the Stanford link: Quantum Entanglement explained by the Stanford Encyclopedia of Philosophy
That is the best explanation available on the web and draws heavily from Shrodinger's original - you cannot do better. Mine is not better. But if that's too long, then, here's mine, a different take.
Let me start with an observation: all the explanations that you typically find when searching for an explanation of entanglement lead to some sort of statement about the existence of things that you can "see" (or observe or measure) and the concept of the "state" of quantum system, usually particles of some kind.
These explanations fail to address the fact that most folks think in terms of "sets" of objects and "observations" as some form of properties in the set. Nothing could be further away and more misleading when it comes to quantum systems since quantum systems are not related to sets of objects but are rather related to the possibilities for describing objects relative to you, their observer. These possibilities are infinite and are described in ways that take those infinities of possibilities into account - they are described by a choice of functions (and that universe of choices by which to choose functions that can describe the objects is itself infinite). Think of a description like this: there are an infinite number of ways to write a description for the number "7" - here are a few:
1. 0 + 7 = 7
2. 3 + 4 = 7
3. 1 + 2 + 4 = 7
4. 0 + 0 + 1 + 6 = 7
5. etc ...
But now I am diving into an area that I will defer so back to entanglement. Some popular descriptions, as I noted, lead you immediately into trouble because the counter-intuitive nature is difficult while thinking of a quantum system like a classical system (which is thought of like a set of things that you can describe with certitude). Quantum systems are spaces of inter-relating entities whose futures are created in part by their past interactions and your choice of how and what you wish to observe. Those choices lead to connected behaviours that Shrodinger called "entangled".
Here are some of the the other popular explanations, for example:
Science Daily Definition of Entanglement
Wikipedia's Definition of Quantum Entanglement
So I offer the following instead: a quantum system is not a set, but, a weird kind of "flow" or "flows". Think of air flow - you cannot see it but a flag waves in the wind. The quantum flows are invisible - in fact, they are so very invisible that even those qualified in the art cannot picture them so whatever I share with you is going to be a hack. Reading my description in the last few sentences, I am already in despair! Let me try again:
Another way to think of it is this: imagine two flags, waving in the wind. Certainly, they will not wave in opposite directions! Their waviness will in some ways appear to be similar, but, not identical. However, if one flag is waving this way, the other flag might be waving that way, and, there may appear to be a kind of regularity. In other words, because we have a the wind, blowing in the same direction, and generally with the same forces, both flags wave in a similar, correlated way. But this is not Quantum. This is Classical. Complicated classical but not quantum.
Quantum entanglement is not like that kind of correlation with the "flows" being a kind of wind of motions and the particles being like the flags. Quantum entanglement is more like the idea that I wanted to see a flag, so I took a bike ride and saw two flags blowing in the wind *because* I wanted to see a flag and took a bike ride --- yes, I know, this sounds weird.
This example or analogy, which it is not of entanglement, is difficult to picture because it is as if I am a participant in creating the outcome - well, actually, I am. That is a quantum feature (my being a participant in creating the outcome).
Here's another one: toss a coin. It can land heads or tails. If I toss the coin and it lands "heads" you know that the other side is "tails" that it has landed on. Imagine now two sheets of paper: one has the words "heads" on one and the other has the word "tails". If I put the papers into two separate envelopes and I ask you to go home with the sealed envelope, you don't know if it has heads or tails written on it. But, if I call you and tell you that my envelope has "heads" then you never need to open your envelope because you have the information and you can infer that the contents of the envelope is "tails". Put another way, imagine you have a coin, say a nickel, and I have a coin, say a quarter and we toss them: every single time, without fail, when your nickel lands on "heads", my quarter just so happens to land on "tails" and vice versa: so it is as if they are invisibly connected but not in any way we can see. If you go home and I go to my home and we repeat the coin tosses, the results are still the same - we can call each other to check on it, after we have tossed the coins and they have landed. Mine lands "heads" and yours is, of course, "tails".
Quantum entanglement is not like this either, but I am trying.
Let me try again: things that you can say get related because of an association or reason might be as disparate as the concept of carrying an umbrella out into sunny skies because the news reported it might rain - this correlation or association between two different things, one of which is observed, you carrying and umbrella, and the other which is hidden, the rain, because the skies are sunny, is like getting closer to the idea of entanglement.
But even this metaphor is not entanglement.
I guess you see where I am going with this. There is a kind of connection that is like a correlation or association but does not really involve anything other than that some information exists. How that information is related to the coin toss, or the umbrella and the rain, is not known - only that it is repeatable and regular in how we can, as outsiders, see the outcomes.
The principle that is strange in entanglement is that of "nonlocality". Imagine our two coins sent with us in opposite directions and the state of one of them is altered (like I turn my quarter to "heads"), the second instantly alters its state (your nickel becomes "tails") in response no matter how far apart the two may be. This physical example is just not realistic but that is what happens at the microscopic level (e.g. photons that are entangled but going in opposite directions).
Albert Einstein he called it "spooky action at a distance" because there was no apparent physical connection underlying the phenomena and no apparent way to explain it.
Some entanglement phenomena are only just emerging such as as association between time and energy: "Now, physicists are unmasking a more fundamental source for the arrow of time: Energy disperses and objects equilibrate, they say, because of the way elementary particles become intertwined when they interact — a strange effect called “quantum entanglement.” ... “Finally, we can understand why a cup of coffee equilibrates in a room,” said Tony Short, a quantum physicist at Bristol. “Entanglement builds up between the state of the coffee cup and the state of the room.” - quoted from Time’s Arrow Traced to Quantum Source
So now we come to the really, really tough part of this blog: how can we represent entanglement?
1. Conceptual Entanglement - we can look at concepts of entanglement for computer systems by looking at the semantics of concept combinations and their representations. A good place to start is Quantum Entanglement in Concept Combinations
2. Representating Entanglement as an artifact in a graph structure: a paper to ideate with is: Entangled graphs: Bipartite entanglement in multi-qubit systems
3. Entanglement polytopes: there are geometries that may prove useful as representational structures and the following two papers should prove sufficient to get you started: Convex polytopes and quantum separability and, the paper here: Entanglement Polytopes
4. Hidden Symmetries, Hidden Variable, Knot Topological and representations of entanglements that are not really "quantum" but are quantum inspired or quasi-quantum (since the work of several physicists has shown that there are no hidden variables): Quantum logic as superbraids of entangled qubit world lines ; Experimental Demonstration of the Topological Phase for Entangled Qubits ; and other references that are simply just way out on the edge (and perhaps a new mathematical model, or physics will be needed to produce a better representation concept for entanglement); and finally (at least on this short list):
5. Using Alternative Probabilistic models such as Quantum Bayesianism to specify models in which entanglement can be represented.
Well, that's all for now, until next time, when we look at Science of Quanta. What is a quantum of information? Is there a quantum of knowledge? Of wisdom? And how could we represent such things, if it is even possible?
Monday, October 6, 2014
Science-I: Structures for Data Representations for Quantum Computing
Science-I: Structures for Data Representations for Quantum Computing and Classical Systems
1. INTRODUCTION
We are going to discuss data representations where we attempt to shed light on how to bridge the gap between systems that are fully Quantum to systems that are Pseudo-Quantum (or Quantum like) to systems that are classical in which data is represented by classical bits but that offer some of the characteristics of Quantum Computing, such as quantization (quantization of data, versus quantization of information versus quantization of knowledge), superposition (read several bits in unit time, or, access multiple facets of information or achieve knowledge access in a way that simultaneously cross-cuts several aspects of knowledge) and entanglement (in which entities are correlated in indirect ways, such as the analog to the way a hologram re-creates the whole image from a fragment of its parts which might be seen as the entanglement of the whole throughout its parts).
I cannot possibly cover all this ground in a single blog. So I am going to do it in several blogs. My aim is to cover the conceptual ground and then bring it to actual implementation ground.
First, the Wikipedia definition:
2. Superposition of States
Wikipedia tells us about on definition of Quantum Superposition:
Quantum Superposition
Quantum superposition is a fundamental principle of quantum mechanics that holds a physical system—such as an electron—exists partly in all its particular theoretically possible states (or, configuration of its properties) simultaneously; but when measured or observed, it gives a result corresponding to only one of the possible configurations (as described in interpretation of quantum mechanics).
This definition presupposes a physical design and a foundation in the axioms of quantum theory. But, let us pause to ask what it is about superposition that is important, as properties, and then return to its concept in order to pave the way for a discussion of design and implementation.
Accord to Quantum Superposition from Princeton:
Quantum superposition refers to the quantum mechanical property of a particle to occupy all of its possible quantum states simultaneously. Due to this property, to completely describe a particle one must include a description of every possible state and the probability of the particle being in that state.
Superposition, from a classical perspective is available through a number of representations that allow us to bijectively represent a collection of states in a single observation. Here are some examples:
(i) Fourier Theory: states are represented by individual sines and the aggregate observation is a waveform ( Fourier Series ).
(ii) Prime Number Theory: states are represented by prime numbers and the aggregate observation is a number that is the sequence of primes multiplied together. In 1679, Leibniz invented his method for representing aggregate concepts in terms of simpler concepts in terms of prime factors. The interested reader can find more here: Leibniz's Characteristica Universalis and also here: Gottfried Leibniz
(iii) Hereditarily Finite Sets: bijective encoding of hereditarily finite sets onto natural numbers where these sets encode the states and the observable is the natural numbers ( Moto-o Takahashi. A Foundation of Finite Mathematics. Publ. Res. Inst., Math. Sci., 12(3):577–708, 1976 ; and see also Paul Tarau. 2010. Declarative modeling of finite mathematics. In Proceedings of the 12th international ACM SIGPLAN symposium on Principles and practice of declarative programming (PPDP '10). ACM, New York, NY, USA, 131-142. DOI=10.1145/1836089.1836107 http://doi.acm.org/10.1145/1836089.1836107)
(iv) Combinatorial Geometric Objects: finite structures that encode the combinations or permuations (aka configurations) of objects as a state: permutohedra, combinohedra and relations between such structures and volumes or angles or surface-areas. For example see "The Amplituhedron".
(v) Multidimensional Venn Diagrams: can also provide a superposition as noted here: especially Multideminsional Venn Diagrams or Venn Diagram of Particle Interactions and also here: Venn
The issue with Venn Diagrams is that we tend to forget that they exits and are useful logical objects.
(vi) Quantum Neural Networks in which a single layer network is trained concurrently with several other single layer network to produce a single state multilayer network. In this case, the superposition occurs through the layering of networks. See for example: Quantum Neural Net and Quantum Inspired Nets.
(vii) Density Matrix: A density matrix is the name given to the operator which is represented as a matrix. This matrix represents a the mixed state as a sum of expectation values of each state weighed by a probability factor. However, while the density matrix represents the distribution of states in terms of probabilities it is not itself the actual superposition: the superposition is given by the computation of the state in terms of the results of the interfering wave function's amplitudes. Therefore, there is a relationship between the superposition (which is seen after the act of measurement) and the probabilities of the states but the point is important to distinguish.
(viii) Phasors: a less well known re-representation for superposition of states is to use a phasor which is essentially the motion of a point on the circumference of a circle driven by the behavior of the wavefunction (as it rises and falls). Phasors are vectors and can be composed or decomposed to produce superpositions of states represented by the driving waves. This dynamical aspect, which is that properties of the medium can be factored into the way the driving signal (wave) is operating is an important point sometimes not well defined in descriptions, for example, and has to be inferred, as here on the Wikipedia page: Phasor
(ix) Wavelets: the superposition (of wavelet pyramids) is used classically in image representations. Wavelets
(x) Twistor Structures: a linear superposition of dynamical processes can be described by different twistor spaces in which the (dynamical) state represents behaviors. For example, in scattering amplitudes (as seen in collisions between particles that then break up into other particles) a positive or negative orientation is assigned to each particle that is either outgoing or incoming. However twistors can, in general represent aribtrary superposition structures of dynamical processes. They are not extensively discussed in contemporary literature as a representation for quantum computing or quantum-like computing processes. To learn more about this fascinating structure, have a look here: Twistor Theory
(xi) R-Functions: an interesting option for representing superpositioning is the use of Rvachev-Functions, an interesting variant of implicit function theory, and one that I myself like to use for its fast computational properties and relation to Boolean logic: Rvachev Function and a for a good primer, here: Primer
(xii) Topology: superposition represented using ideas in knot-theory and topology. This is another exotic model for superpositioning of states and one that also has physical promise. For the curious see here: Computing with Quantum Knots and here for a more advanced version: Topological Quantum Computing or for an Arvix paper: Quantum Knots
There are likely several more ways to encode a superposition of states, and, I will try to add to the methods.
However, what I would like to draw attention to is that it is not any one method for superposition representation but that the method connects with other concepts, like entanglement, in order to become useful in the quantum inspired models of computing.
More next time, in Science-II.
1. INTRODUCTION
We are going to discuss data representations where we attempt to shed light on how to bridge the gap between systems that are fully Quantum to systems that are Pseudo-Quantum (or Quantum like) to systems that are classical in which data is represented by classical bits but that offer some of the characteristics of Quantum Computing, such as quantization (quantization of data, versus quantization of information versus quantization of knowledge), superposition (read several bits in unit time, or, access multiple facets of information or achieve knowledge access in a way that simultaneously cross-cuts several aspects of knowledge) and entanglement (in which entities are correlated in indirect ways, such as the analog to the way a hologram re-creates the whole image from a fragment of its parts which might be seen as the entanglement of the whole throughout its parts).
I cannot possibly cover all this ground in a single blog. So I am going to do it in several blogs. My aim is to cover the conceptual ground and then bring it to actual implementation ground.
First, the Wikipedia definition:
A quantum computer is a computation system that makes direct use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data..We'll start with the concept of superposition.
2. Superposition of States
Wikipedia tells us about on definition of Quantum Superposition:
Quantum Superposition
Quantum superposition is a fundamental principle of quantum mechanics that holds a physical system—such as an electron—exists partly in all its particular theoretically possible states (or, configuration of its properties) simultaneously; but when measured or observed, it gives a result corresponding to only one of the possible configurations (as described in interpretation of quantum mechanics).
This definition presupposes a physical design and a foundation in the axioms of quantum theory. But, let us pause to ask what it is about superposition that is important, as properties, and then return to its concept in order to pave the way for a discussion of design and implementation.
Accord to Quantum Superposition from Princeton:
Quantum superposition refers to the quantum mechanical property of a particle to occupy all of its possible quantum states simultaneously. Due to this property, to completely describe a particle one must include a description of every possible state and the probability of the particle being in that state.
Superposition, from a classical perspective is available through a number of representations that allow us to bijectively represent a collection of states in a single observation. Here are some examples:
(i) Fourier Theory: states are represented by individual sines and the aggregate observation is a waveform ( Fourier Series ).
(ii) Prime Number Theory: states are represented by prime numbers and the aggregate observation is a number that is the sequence of primes multiplied together. In 1679, Leibniz invented his method for representing aggregate concepts in terms of simpler concepts in terms of prime factors. The interested reader can find more here: Leibniz's Characteristica Universalis and also here: Gottfried Leibniz
(iii) Hereditarily Finite Sets: bijective encoding of hereditarily finite sets onto natural numbers where these sets encode the states and the observable is the natural numbers ( Moto-o Takahashi. A Foundation of Finite Mathematics. Publ. Res. Inst., Math. Sci., 12(3):577–708, 1976 ; and see also Paul Tarau. 2010. Declarative modeling of finite mathematics. In Proceedings of the 12th international ACM SIGPLAN symposium on Principles and practice of declarative programming (PPDP '10). ACM, New York, NY, USA, 131-142. DOI=10.1145/1836089.1836107 http://doi.acm.org/10.1145/1836089.1836107)
(iv) Combinatorial Geometric Objects: finite structures that encode the combinations or permuations (aka configurations) of objects as a state: permutohedra, combinohedra and relations between such structures and volumes or angles or surface-areas. For example see "The Amplituhedron".
(v) Multidimensional Venn Diagrams: can also provide a superposition as noted here: especially Multideminsional Venn Diagrams or Venn Diagram of Particle Interactions and also here: Venn
The issue with Venn Diagrams is that we tend to forget that they exits and are useful logical objects.
(vi) Quantum Neural Networks in which a single layer network is trained concurrently with several other single layer network to produce a single state multilayer network. In this case, the superposition occurs through the layering of networks. See for example: Quantum Neural Net and Quantum Inspired Nets.
(vii) Density Matrix: A density matrix is the name given to the operator which is represented as a matrix. This matrix represents a the mixed state as a sum of expectation values of each state weighed by a probability factor. However, while the density matrix represents the distribution of states in terms of probabilities it is not itself the actual superposition: the superposition is given by the computation of the state in terms of the results of the interfering wave function's amplitudes. Therefore, there is a relationship between the superposition (which is seen after the act of measurement) and the probabilities of the states but the point is important to distinguish.
(viii) Phasors: a less well known re-representation for superposition of states is to use a phasor which is essentially the motion of a point on the circumference of a circle driven by the behavior of the wavefunction (as it rises and falls). Phasors are vectors and can be composed or decomposed to produce superpositions of states represented by the driving waves. This dynamical aspect, which is that properties of the medium can be factored into the way the driving signal (wave) is operating is an important point sometimes not well defined in descriptions, for example, and has to be inferred, as here on the Wikipedia page: Phasor
(ix) Wavelets: the superposition (of wavelet pyramids) is used classically in image representations. Wavelets
(x) Twistor Structures: a linear superposition of dynamical processes can be described by different twistor spaces in which the (dynamical) state represents behaviors. For example, in scattering amplitudes (as seen in collisions between particles that then break up into other particles) a positive or negative orientation is assigned to each particle that is either outgoing or incoming. However twistors can, in general represent aribtrary superposition structures of dynamical processes. They are not extensively discussed in contemporary literature as a representation for quantum computing or quantum-like computing processes. To learn more about this fascinating structure, have a look here: Twistor Theory
(xi) R-Functions: an interesting option for representing superpositioning is the use of Rvachev-Functions, an interesting variant of implicit function theory, and one that I myself like to use for its fast computational properties and relation to Boolean logic: Rvachev Function and a for a good primer, here: Primer
(xii) Topology: superposition represented using ideas in knot-theory and topology. This is another exotic model for superpositioning of states and one that also has physical promise. For the curious see here: Computing with Quantum Knots and here for a more advanced version: Topological Quantum Computing or for an Arvix paper: Quantum Knots
There are likely several more ways to encode a superposition of states, and, I will try to add to the methods.
However, what I would like to draw attention to is that it is not any one method for superposition representation but that the method connects with other concepts, like entanglement, in order to become useful in the quantum inspired models of computing.
More next time, in Science-II.
Friday, October 3, 2014
Philosophy Before Formality: Some Definitions For Quantum AI
Philosophy Before Formality: Some Definitions For Quantum AI
"Mathematicians are like Frenchmen: whatever you say
0. PREFATORY NOTES
It is already very difficult to follow almost any proof in mathematical texts without substantial background, or inline, though extraneous comments and explanations just like it is difficult to follow someone else's uncommented computer code: it can be done, but, it just takes a lot longer for no good reason.
William Kingdon Clifford said it like this in Common Sense in the Exact Sciences (1885), 21.
" We may always depend on it that algebra, which cannot be translated into good English and sound common sense, is bad algebra."
So the idea in these blogs is to first express the philosophy, the idea and then formulate that in mathematical expression. The goal is to state things in reasonable clear, simple and concise English before stating those things with equations.
What I am discovering for myself is that in writing this blog, I am revealing to myself my own mental machinery and its gaps, holes and deficiencies: I hope those that read this will do the same and point out where I am being ambiguous or downright useless.
1. INTRODUCTION
I am going to define what I mean by Quantum Computing Artificial Intelligence (QCAI).
My definition is the not the right definition: it is my point of view and I am writing exposition to clarify my point of view. I welcome other viewpoints.
"The real danger is not that computers will begin to think like men,
We seem to have descended closer to the way computers communicate, using short abbreviated codes in tweets, adjusting our mind space in order to search with specific combinations keywords: the Search engines have us well trained!
QCAI in my view should contribute to bringing computers closer to the level at which the human operates and not diminish the human down to the level of a machine. Yet software trains us. We do not train software. We buy an app and we have to "learn" it. It does not learn us.
2. Quantum Computing Artificial Intelligence (QCAI)
Quantum AI is concerned pattern discovery which then becomes the basis for pattern recognition. Traditional computing is concerned, for the most part, with pattern-recognition. Usually parameters of recognition become statistical entities used later in discovery but these parameters also box-in what is possible --- nothing outside the realm of the parameters can be discovered. Quantum AI focuses on the discovery of patterns, chains of patterns, and transformations of patterns utilizing the fundamental concepts essential to the identity of what we are called Quantum AI. Patterns discovered are then learned and used in pattern recognition.
The words “artificial intelligence” were coined renowned computer scientist John McCarthy, in 1956 at the first conference in Dartmouth. So defines the subject? I will defer and refer the reader to the pages at Stanford from the great man himself:
WHAT IS ARTIFICIAL INTELLIGENCE?, by John McCarthy
http://www-formal.stanford.edu/jmc/whatisai/whatisai.html
One part of the document in FAQ is the following quote:
"Q. Are computers fast enough to be intelligent?
A. Some people think much faster computers are required as well as new ideas. My own opinion is that the computers of 30 years ago were fast enough if only we knew how to program them."
I underlined and italicized the words because our paradigm of programming AI has relied on classical physics.
So, the objective with QCAI is to answer the question "if only we knew how to program them".
So, here is my working definition:
Quantum Computing Artificial Intelligence, or Quantum Artificial Intelligence is the science, design and implementation of intelligent computer programs that have competency and performance at the level or beyond of human intelligence in complex understanding, value judgements, reasoning, and decision making.
So, clearly, this definition embodies the Science, the Design and the Implementation aspects. For the next few blog entries we will focus on the Science, then the design and then the implementation.
3. CAVEATS
Here are all my caveats, as better written than I, by others I admire:
How to be a Bad Theoretical Physicist
http://www.staff.science.uu.nl/~hooft101/theoristbad.html
Are you a Quack?
http://insti.physics.sunysb.edu/~siegel/quack.html
The Crackpot Index
http://math.ucr.edu/home/baez/crackpot.html
How to be a Good Theoretical Physicist
http://www.staff.science.uu.nl/~Gadda001/goodtheorist/
4. POPULAR LINKS
And, to get familiar with some of the emerging sites and research:
http://www.nas.nasa.gov/quantum/research.html
"One of the central open questions in the field of quantum computing is the existence of efficient quantum heuristic algorithms for solving classically intractable instances of combinatorial optimization problems that are found at the core of many of NASA’s missions." - quoted from NASA
http://www.usra.edu/quantum/rfp/
"The Universities Space Research Association (USRA) is pleased to invite proposals for Cycle 1 of the Quantum Artificial Intelligence Laboratory Research Opportunity, which will allocate computer time for research projects to be run on the D-Wave System at NASA Ames Research Center (ARC) for the time period November 2014 through September 2015."
https://plus.google.com/+QuantumAILab/posts
"The Quantum Artificial Intelligence Lab is a collaboration between Google, NASA Ames Research Center and USRA. We're studying the application of quantum optimization to difficult problems in Artificial Intelligence."
5. UNTIL NEXT TIME
It's all about the Science.
"Mathematicians are like Frenchmen: whatever you say
to them they translate into their own language and
forthwith it is something entirely different."
-- Johann Wolfgang von Goethe
0. PREFATORY NOTESIt is already very difficult to follow almost any proof in mathematical texts without substantial background, or inline, though extraneous comments and explanations just like it is difficult to follow someone else's uncommented computer code: it can be done, but, it just takes a lot longer for no good reason.
William Kingdon Clifford said it like this in Common Sense in the Exact Sciences (1885), 21.
" We may always depend on it that algebra, which cannot be translated into good English and sound common sense, is bad algebra."
So the idea in these blogs is to first express the philosophy, the idea and then formulate that in mathematical expression. The goal is to state things in reasonable clear, simple and concise English before stating those things with equations.
What I am discovering for myself is that in writing this blog, I am revealing to myself my own mental machinery and its gaps, holes and deficiencies: I hope those that read this will do the same and point out where I am being ambiguous or downright useless.
1. INTRODUCTION
I am going to define what I mean by Quantum Computing Artificial Intelligence (QCAI).
My definition is the not the right definition: it is my point of view and I am writing exposition to clarify my point of view. I welcome other viewpoints.
"The real danger is not that computers will begin to think like men,
but that men will begin to think like computers."
-- Sydney J. Harris
We seem to have descended closer to the way computers communicate, using short abbreviated codes in tweets, adjusting our mind space in order to search with specific combinations keywords: the Search engines have us well trained!QCAI in my view should contribute to bringing computers closer to the level at which the human operates and not diminish the human down to the level of a machine. Yet software trains us. We do not train software. We buy an app and we have to "learn" it. It does not learn us.
2. Quantum Computing Artificial Intelligence (QCAI)
Quantum AI is concerned pattern discovery which then becomes the basis for pattern recognition. Traditional computing is concerned, for the most part, with pattern-recognition. Usually parameters of recognition become statistical entities used later in discovery but these parameters also box-in what is possible --- nothing outside the realm of the parameters can be discovered. Quantum AI focuses on the discovery of patterns, chains of patterns, and transformations of patterns utilizing the fundamental concepts essential to the identity of what we are called Quantum AI. Patterns discovered are then learned and used in pattern recognition.
The words “artificial intelligence” were coined renowned computer scientist John McCarthy, in 1956 at the first conference in Dartmouth. So defines the subject? I will defer and refer the reader to the pages at Stanford from the great man himself:
WHAT IS ARTIFICIAL INTELLIGENCE?, by John McCarthy
http://www-formal.stanford.edu/jmc/whatisai/whatisai.html
One part of the document in FAQ is the following quote:
"Q. Are computers fast enough to be intelligent?
A. Some people think much faster computers are required as well as new ideas. My own opinion is that the computers of 30 years ago were fast enough if only we knew how to program them."
I underlined and italicized the words because our paradigm of programming AI has relied on classical physics.
So, the objective with QCAI is to answer the question "if only we knew how to program them".
So, here is my working definition:
Quantum Computing Artificial Intelligence, or Quantum Artificial Intelligence is the science, design and implementation of intelligent computer programs that have competency and performance at the level or beyond of human intelligence in complex understanding, value judgements, reasoning, and decision making.
So, clearly, this definition embodies the Science, the Design and the Implementation aspects. For the next few blog entries we will focus on the Science, then the design and then the implementation.
3. CAVEATS
Here are all my caveats, as better written than I, by others I admire:
How to be a Bad Theoretical Physicist
http://www.staff.science.uu.nl/~hooft101/theoristbad.html
Are you a Quack?
http://insti.physics.sunysb.edu/~siegel/quack.html
The Crackpot Index
http://math.ucr.edu/home/baez/crackpot.html
How to be a Good Theoretical Physicist
http://www.staff.science.uu.nl/~Gadda001/goodtheorist/
4. POPULAR LINKS
And, to get familiar with some of the emerging sites and research:
http://www.nas.nasa.gov/quantum/research.html
"One of the central open questions in the field of quantum computing is the existence of efficient quantum heuristic algorithms for solving classically intractable instances of combinatorial optimization problems that are found at the core of many of NASA’s missions." - quoted from NASA
http://www.usra.edu/quantum/rfp/
"The Universities Space Research Association (USRA) is pleased to invite proposals for Cycle 1 of the Quantum Artificial Intelligence Laboratory Research Opportunity, which will allocate computer time for research projects to be run on the D-Wave System at NASA Ames Research Center (ARC) for the time period November 2014 through September 2015."
https://plus.google.com/+QuantumAILab/posts
"The Quantum Artificial Intelligence Lab is a collaboration between Google, NASA Ames Research Center and USRA. We're studying the application of quantum optimization to difficult problems in Artificial Intelligence."
5. UNTIL NEXT TIME
It's all about the Science.
Thursday, October 2, 2014
Quantum and Pseudo-Quantum Computing
Quantum and Pseudo-Quantum Computing.
Well, well, well - I guess I finally caught the drift into deeper waters despite my own shallow sailing! This is my very first Blog. It's just an introduction about what I am interested in and things I would like to share. Here goes:
Quantum computing for Artificial Intelligence (QCAI) sounds so lofty and pretentious that I am almost scared to post on this. I guess, as far as classical systems go (like this Intel Core-I7 Samsung laptop) I should really call it inspired by quantum or Pseudo-Quantum (PQCAI).
I will explore some of the options for using the concepts from Quantum Computing to enable Pseudo-Quantum algorithms to be written for real world AI jobs: the main idea is that we start with the fundamental concepts of quantum computing and AI: from these, we delve into what we can achieve with various ways of combining or being inspired by the concepts to write real world software.
I am not that interested in theoretical discussions but I am interested in actual performance of applications.
Here are some thoughts:
1. Quantum Science has several variants and flavors in terms of theory and interpretation: Bohm's Pilot Wave, or Copenhagen ... we'll get into this later. What does this have to do with AI from a practical viewpoint?
2. There is a concept of multiple possibilities that are ready to produce an answer: some folks call this wave function collapse and it also has the moniker of Quantum Decoherence ... we'll get into that later too. And, what does it mean for the way we design algorithms in PQCAI?
3. Of course, what discussion could go forward with getting all entangled with the concept of Entanglement itself, which is sometimes very, very loosely referred to as a quantum correlation (bad choice of words here!).
4. And then there is that whole business about things coming in discrete packets or Quanta which is at the core of the physics of Quantum phenomena; and,
5. Lastly, the notion that measurements change the system (or put another way, that interaction is the means to computation). This is a bit off track and more really about my own viewpoint than dogma.
Well, that's plenty for my first cut at attempting to blog: until next time!
Well, well, well - I guess I finally caught the drift into deeper waters despite my own shallow sailing! This is my very first Blog. It's just an introduction about what I am interested in and things I would like to share. Here goes:
Quantum computing for Artificial Intelligence (QCAI) sounds so lofty and pretentious that I am almost scared to post on this. I guess, as far as classical systems go (like this Intel Core-I7 Samsung laptop) I should really call it inspired by quantum or Pseudo-Quantum (PQCAI).
I will explore some of the options for using the concepts from Quantum Computing to enable Pseudo-Quantum algorithms to be written for real world AI jobs: the main idea is that we start with the fundamental concepts of quantum computing and AI: from these, we delve into what we can achieve with various ways of combining or being inspired by the concepts to write real world software.
I am not that interested in theoretical discussions but I am interested in actual performance of applications.
Here are some thoughts:
1. Quantum Science has several variants and flavors in terms of theory and interpretation: Bohm's Pilot Wave, or Copenhagen ... we'll get into this later. What does this have to do with AI from a practical viewpoint?
2. There is a concept of multiple possibilities that are ready to produce an answer: some folks call this wave function collapse and it also has the moniker of Quantum Decoherence ... we'll get into that later too. And, what does it mean for the way we design algorithms in PQCAI?
3. Of course, what discussion could go forward with getting all entangled with the concept of Entanglement itself, which is sometimes very, very loosely referred to as a quantum correlation (bad choice of words here!).
4. And then there is that whole business about things coming in discrete packets or Quanta which is at the core of the physics of Quantum phenomena; and,
5. Lastly, the notion that measurements change the system (or put another way, that interaction is the means to computation). This is a bit off track and more really about my own viewpoint than dogma.
Well, that's plenty for my first cut at attempting to blog: until next time!
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