Data and Algorithm Part-1: Quantum Inspired System Semantics (QUISS) - Overview

 

Introduction

The next set of posts will introduce several elements that will build a mindset, or mental model and a point of view about quantum inspired system semantics that will focus on the actual path to building operational quantum-like data representations, processing and algorithms.

Quantum Inspired System Semantics or QUISS for short follows the KISS principle.  That is, I will do my best to get to the point quickly and efficiently.  However, let's not forget what Feynman and Nielsen said about trying to explain quantum computing or other challenging emerging science.  If it could be written and explained to everyone or anyone, then it would not be new but commonplace and old-hat.

In terms of the approach, we will take things starting at the beginning and provide some concretely usable concepts, in several cases, implementations as well, and example data input/outputs and value.

Bits, Bytes and the Qubit

 

Traditional computing uses the familiar abstraction of the Bit over its implementation as an electrical circuit, or even a mechanical circuit (like Turing's original designs) whose voltage or mechanical lever is either above or below a certain threshold.

The concept of aggregating a number of bits, in order, using the binary base numbering system was all that was needed to complete the most basic and fundamental data structure:  the byte.  It is on bits and bytes that all of modern computing rests.

From a linguistic perspective it was natural, therefore, that a name that had as its past, the word "bit" in it would be chosen as the simplest term for a Quantum datum.   But unlike the word "Bit", which is crisp clear and simple, with an underlying abstraction built on electrical or mechanical circuits, adding the word "Quantum" changes everything so much to the point that the word "QuBit" leads one into a morass.

The best explanation is not the Wiki Qubit, but the David Deutch article on "It from Bit".

A Qubit is neither crisp, clear nor simple: it is a superposition of states of objects representing states, such as state vectors whose elements are some other sorts of mathematical objects that have nothing to do with voltage thresholds or the positions of mechanical levers.

In fact, the Qubit concept as you find it online incorporates two notions:  the first is that there is a probability scale between the elements that make up a qubit and second that knowledge of the first element automatically implies knowledge of a second.  So, as an example,  consider the question whether if it will rain tonight: instead of a classical "yes" / "no" stated as binary {1} or {0}, you have some range from [0, 1] representing the probability of the outcome that it will rain and not rain.  In other words, the underlying elements are not probabilities but wavefunctions whose amplitudes when subjected to squaring eliminate the complex plane to produce a real value that is then interpreted as the probability.

The issue is that the underlying abstraction for a qubit is not a circuit but a complex vector space, and then, the underlying abstraction of that complex vector space is the mental model of a quantum entity as inferred from field theory and other branches of physics.

A quantum observable is neither a real variable, like a classical degree of freedom, or probability measurement nor a discrete variable like a classical bit, or a random choice, but a much more fascinating, and deeply significant object that entangles and integrates all the discrete and continuous aspects in all the dimensional as well as non-dimensional but measurable phenomena.

The implication is actually staggering:  how does a quantum entity make the transition from one state to another?  Does it just "jump" between states? The answer is incredible and given by quantum theory is that it makes it
continuously but is only observable discretely!

The point here is that the Qubit is nothing like the Bit and not even remotely so.  It would have, in my opinion, been better to chose some other completely different name to avoid the confusion that this now produces. 

Qubits, Qutrits, Qudits

The key goal in representation is to figure what data structures are most useful and enable suporpositioning, quantization and entanglement.  If we take a step back to see the forest from the trees, we realize the combinatorial structures and geometries have a utility all their own:  Alexander Grothedieck developed a theory of "places", formally called Topose Theory, as a replacement and as an advancement of the theory of categories where the intuitive patterns of geometry could be localized and captured in model for reuse.

The major idea we are presenting here is that geometry is intimately tied into combinatorics which is the key to the quantum probabilistic description of information structure.
 

[TO BE CONTINUED]

 

 

Quantum Information and Quantum Of Information - III

Review from last post:

In getting to what is a Quantum of Information, I discussed some notions around ontology synthesis which is at the core of understanding any domain:  the ability to conceptualize the world and domain in it. 

I then went on to say that when a given hypothesized theory T2 contains more information than a competing theory T1, that we can find the distribution where the Quantum probability density matrix that defines the ontology states can be discovered and refined with increasing precision driven by using knowledge contained in theory T2 over that contained in T1.

The strategy used is called Quantum Inspired Semantic Tomography (QIST) drawn from Quantum State Tomography but adapted for information science.
High quality decision making depends on a high utility of knowledge which in turn is based on identifying the structure and the probability distribution of variants, the importance of which can only be lifted from raw data itself or the evidence about the data.

Finally, the subject at hand: The Quantum Of Information

Infon:  A Quantum Of Information

 

So, and here is the critical thing:  the traditional model of thinking in AI in general is based on the Symbol Hypothesis much like physics had originally been conceived of in terms of physical particles when, in fact, particles are just the observable and measurable phenomena of fields. In other words, particles emerge as the kinks of a field, such as the electromagnetic field.  The Physical Symbol System Hypothesis was first put forth over half a century ago in their paper,  Computer science as empirical inquiry: symbols and search,  for which they won the prestigious Turing Award.  The theory, rooted firmly in the foundations of Atomism, has been the cornerstone for intelligent systems.

For a contemporary review of the status of the symbol hypothesis, see for example the Stanford paper by Nils J. Nilsson.

I am not going to go further into this except to state, categorically, that it becomes obvious that whatever symbol system emerges that it must be ontologically grounded and that it must have some relationship to fidelity, accuracy and precision to the concepts that it seeks to express.

The central question, therefore, for Quantum AI is not about the validity or invalidity of the symbol hypothesis, just as atoms, electrons, protons and all the other more exotic particles in the atomistic tradition are not invalid but rather, how can symbols themselves be created such that they, as Pierce's semiotic elements, are fit to the purposes of their environment (which includes the entities that utilize them as well as their significance with respect to the environment in which the entities exist).

The concept is the Quantum Of Information:  how is a symbol to be rationally existent with a utility, preference and objective value in semiosis.

My hypothesis is that the concept of the Infon has most of the character of the quantum of information but does not explain how to create such an infon without the presence of a human author.

Our goal is that the machine is the author of concepts and teaches us about them.

But, in order to achieve this, therefore, we must identify, just as physicists have done, what are the raw mathematical materials to use in developing models that produce the observable and usable infons (like the Higgs that give mass to particles and the quarks and gluons that give cohesion to the nuclei of atoms).

If indeed we are to continue to adopt and to use the atomistic model, we must also acknowledge that there is the possibility that the atomistic model has a deeper structure underlying it.

It is this deeper structure that I believe Quantum inspired, pseudo-Quantum and alternative thinking can model.

Implementing Infons 

 

In the next post, I will formally introduce a field model out of which such a structure could possibly be defined as McLennan states:  "A field computer processes fields, that is, spatially continuous arrays of continuous value, or discrete arrays of data that are sufficiently large that they may be treated mathematically as though they are spatially continuous. Field computers can operate in discrete time, like conventional digital computers, or in continuous time like analog computers."

In recent news, new engineering prototypes such as the Metasurface analog computation substrate my provide the bridge between quantum-like analog based computing for addressing many of the ideas in these blogs.

Computer scientists believe quantum computers can solve problems that are intractable for conventional computers because they work according to principles that can solve problems whose solution will never be feasible on a conventional computer. No one is advocating that the Church-Turing thesis is violated, but, that in the linear sequential Turing machine of the classic kind that this model itself limits what is possible:  the Turing machine comes in many different flavors and in another blog I will address these issues as I have so far never been happy with the dispersed fragments on the subject that I have had to collect.

In fact, Michael Nielsen writes that many folk ask for a simple concrete explanation of a quantum computer.  Feynmann himself was asked for a simple concrete explanation of why he won a Nobel prize by a newsman.

Nielsen's answer:  " The right answer to such requests is that quantum computers cannot be explained in simple concrete terms; if they could be, quantum computers could be directly simulated on conventional computers, and quantum computing would offer no advantage over such computers. In fact, what is truly interesting about quantum computers is understanding the nature of this gap between our ability to give a simple concrete explanation and what’s really going on"

Feynmann's answer to his question: "Hell, if I could explain it to the average person, it wouldn't have been worth the Nobel prize."

An Analogical Paradigm of Representation

I assume that the brain uses an analog model with an analogical pattern strategy as a means to representation, the whole resting on a foundation of quantum mechanics: in other words, an in the light of Peircean semiotics, that what is perceived emerges as the result of the interactions between primordial analogs for patterns.
The implication is that objects are represented in the brain using ephemeral infons that are spatial analogs of them that synthesize mimicking models of objects:   it seems to us only because we have never seen those objects in their raw form, inside our own minds, but only through our perceptual representations of them that they are indeed the objects themselves and not their proxies.

Perception is usually overlooked in reasoning because the the illusion is that the objects of reasoning exists when in fact they are interpretants in the mind of the beholder, like the virtual particals in quantum physical computations.

The world of perception appears so much like the real world of which it is merely a copy that we forget that there is actually a significant distinction.

In other words, an abstract symbolic code, as suggested in the symbol hypothesis paradigm, is not used to represent anything in the brain; nor are they encoded by the activation of individual cells or groups of cells representing particular features detected in the scene, as suggested in the neural network or feature detection paradigm.  Rather, these abstractions exist as structures on the substrate of the physico bio electrical structures that sustain their existence.

The reason why the brain expresses perceptual experience in explicit spatial form originates in evolution where a brain capable of processing that spatial information provided higher survivability (i,e. to jump and climb into trees). In fact the nature of those spatial algorithms is itself open to investigation.

These spatial and sensori-motor structures produce the schemes which can support analogical representation and reasoning.

Quantum Information and Quantum Of Information - II

Five Working Assumptions:

The goal in this blog is the following: to help understanding of the quantum of information by using an ontology engineering example idea.  Building an ontology from a blank-slate is the most human-intensive task today. We would like machines to do this.   The notion of interaction based computing, that underlies the quantum metaphor,  is to build theories using an ontology induced from raw data and then to reconstruct the hitherto unknown ontology from theories abduced from that data and back-tested using a criterion of informativeness: the most informative theories will reconstruct a probability density matrix that represents the most precise ontology states with the best utility for decision making or reasoning.

Earlier, we introduced a few concepts about what you know, or don't know and the idea of interaction.  Now, I would like to make a commitment to a working set of assumptions about how I think about what Quantum Artificial Intelligence (AI) is.  The objective, in me personal viewpoint, of Quantum AI, is to enable computers to function at a higher level of value in the knowledge production chain: that is, to reach towards creativity and innovation.

Quantum AI will need to perform the following five core functions:

1.  Analysis:  ability to process, re-represent and learn from data
2.  Synthesis: transform date into knowledge and then into re-usable wisdom
3.  Context:  hypotheses of plausible futures as interpretations with least information "on the fly".
4.  Fusion: represent higher-order concepts (or ideas) from different, possibly incommensurate semiotic paradigms into a single unified paradigm.
5.  Consciousness: self-awareness and then beyond the “self” the ability to explore higher-order thinking within a "meta-self".

What computational models and representations could we use and how can a system reach towards creativity and innovate itself?  Is that possible or would it violate Godel's Incompleteness Theorem?  Is this still Turing Computation - is it a Turing machine.

The short answer is that no, due to the nature of computation by interaction it would not violate Godel's theorem and, yes, this is actually still Turing machine computation --- to be precise, it fits the model of a Turing Choice Machine.

Let's attend to the the Quantum of Information, which underlies the first point about data "re-representation":  note, I did not say data representation, but rather, re-representation, to encourage thinking about the representation itself as an interactive coming about of how data becomes usable and what its minimal bounds must be in order to be useful (from a Quantum AI point of view).

In order to ground the ideas that I am presenting, let me start by stating that whatever we as humans exchange as thoughts, and that also whatever any sort of intelligence has to exchange in order to express itself must somehow be conveyed by having some concrete statement to make.  Let us take as an axiom of our system that First Order Predicate Logic is the modality of that expression and that this can be rewritten in natural languages (Arabic, French, Chinese, English or Ancient Egyptian) as well as computer languages (C, Prolog, COBOL, Java, etc...).

Of course, we are not making the statement that the mental representation is first-order logic, but only that statements made by the mental machinery can be rewritten in some variant of (modal) predicate logic.

Therefore, we can now set the stage for a preliminary discussion of state-vectors:

1) That the intelligent machinery has some kind of internal state vector or state representation; and,
2) That the output of the mental machinery must serve its basic survival which means that it ought to behave efficiently in the world; and,
3) That the mental machinery approximates the state vector of the world by making good-enough  approximations of the true probability distribution of its
states; so that,
4) Good decisions can be made; and,
5) Future survivability is increased by taking utilitarian actions.

Therefore, from a mathematical perspective, the minimal number of interactions that optimize informativeness of the exchanges, which in turn implies well formed, rich statements, then the higher the fidelity of the approximation of the true state of the world and the better the outcomes (from decision making).

An interaction, seen as the exchange of a logical statement, will be valid with some probability between 0 and 1:  this leads to a the notion of refinements of the interactions so that the each current estimate of the world-state is updated by a rewriting the statements to augment the estimates.

Now we can write this as a mathematical model (of course, as you can see, I have started to sneak in the ideas of Minimum Message length as well its complementary idea of Minimum Interaction distance).

In simple terms, a formula in first order predicate logic (FOPL) will have a level of informativeness that, through an interaction will be rewritten to the point that it represents a close enough approximation to the true state of the world, or at least, that part that is consistently observable of the true state vector.

Therefore, the question becomes what is that difference between the initial proposition in FOPL and the smallest rewrite step forward as a quantum of information that transforms informativeness towards the true state? And what can be used to measure it?

Here is where we will gingerly suggest an approach with some testable hypotheses - not as a right answer (as I don't know the answer, yet) but as a place to begin, and, as a placeholder for a preliminary model that is executable.

For the sake of discourse, let us call our intelligent machinery an agent.

The agent will create a statement about the world and let us call that statement a theory.   We are justified in using the word theory as it implies some kind of ontology and also that the agent has not actually gotten a full understanding of the totality of the world, but theorizes some understanding of some part of the world.

The agent, obviously, needs to test the theory:  in this case, the agent conducts an experiment: doesn't this sound just like the Scientific Method?.    Well, my intention is to draw attention to it and to the work of Charles S. Peirce and his pragmatism of inquiry.

If the agent, as a result of the experiment, which could take the form of a conversation with another agent, is able to rewrite the original theory by enriching its information, then, the agent can produce a new theory (which is usually, though not always, a modification of the original theory).

A simple way to progress is to think in terms of adding one new relation at a minimum or, instead, one new concept and relation to the existent theory, T1, to produce a new theory, T2 where the informativeness of T2 is strictly greater than T1.

The temptation is to linearize thinking and to assume that the steps are somehow linear.  They are not.  The reason is that we do not know what form the steps take.  One notion is that adding a new relationship would at most provide expand the space by a factorial in the number of concepts related by the relationship and the combinations of the relationships in concert with the concepts would add another increase to the descriptive or informative power.

If we take a distinctly Quantum approach and we liken the ur-elements (atoms or literals) of some ontology with a probability distribution calculable in terms of a complex probability amplitude then make the statement:  let Ω be an ontology with probability amplitude distributions amongst its atoms.  Furthermore let the complex plane determine the order structure and the real plane determine the conceptual association structure so that a given theory is defined by the ordered associations between concepts from the ontology.

It should be clear that using this approach, that the two theories T2 and T1 can be distinguished by a divergence measure which we could justifiably state as being in inverse proportion to the informativeness increase from T1 to T2.

Candidates for such measures include quantum Jensen-Shannon, Renyi Relative Entropy, Kullback-Liebler, and Bregman divergence measures.

The way I think about this is that an atom of the ontology is represented as a virtual particle that is defined by its wavefunction ψ(x).   The quantum state corresponds to the configuration in which the ontology is meaningfully constructed such that the Theories T1 and T2 are expressible.

However,  in an unknown quantum state (i.e. we do not know the ontological structure) we are given data for which we need to determine a particle in an unknown quantum state as well as to measure observables from which to induce its wavefunction ψ(x):  here is an example in which quantum inspired views can work forwards and backwards.

But the problem is that we do not know what bases are appropriate so we have to try out several - hence, from these efforts we can determine the wavefunctions and compute the density matrix which then tells us everything we need to know.

The key here is that we must consider that the different roles of the measures correspond to different roles of the particles play in the definition of the ontology and only when truly new particles contribute to state information (i.e non degenerate copies and measurements) then only can we reconstruct the state and from this compute the divergence between states and therefore the informativeness between T1 (made up of some configuration of particles) and T2.

If your heads hurts - well, this is because where we are headed with this blog is Quantum Tomography.

I offer instead, the concept of Quantum Inspired Semantic Tomography:  that is the use of quantum mathematical and operational methods to model and execute ontology synthesis from raw data.

In this sense, ontology synthesis from data is the analog to reconstruct a quantum state from experimental data inputs and this in turn enables theories to be produced and driven by their informativeness computed via the divergence measures.

To put it another way, in order to synthesize the ontology which is the understanding of a domain or the world, a quantum system model which is self-emergent is needed.  That is, the quantum system must correspond to seperate interacting quantum systems that produce theories about the world subject to revision via divergence considerations until a good-enough result is achieved where further quantum effects make no tangible difference to the theory revision.

Mathematically, we could call this a fixed-point.

If we measure the quantized "jumps" between divergences according to some optimality criterion (and I know I have not answered what the optimality is) then we would have a profile of ontological development for any given domain and know where we can make the biggest difference to the biggest gaps within that domain.

Therefore, the theory that produces the largest informativeness from T1 to T2 allows one to reconstruct an unknown distribution of states characterizing an ontology as determined by the probability density matrix.  This is the Quantum Tomography part of the semantics.

At present, this process is done by many talented humans.

But quantum inspired methods might do this too!

Until next time.