Quantum Information Dynamics and Open World Science

One of the fundamental insights of quantum mechanics is that complete knowledge of the state of a quantum system is not possible. Such incomplete knowledge of a physical system is the norm rather than the exception. This is becoming increasingly apparent as we apply scientific methods to increasingly complex situations. Empirically intensive disciplines in the biological, human, and geosciences all operate in situations where valid conclusions must be drawn, but deductive completeness is impossible. This paper argues that such situations are emerging examples of {it Open World} Science. In this paradigm, scientific models are known to be acting with incomplete information. Open World models acknowledge their incompleteness, and respond positively when new information becomes available. Many methods for creating Open World models have been explored analytically in quantitative disciplines such as statistics, and the increasingly mature area of machine learning. This paper examines the role of quantum theory and quantum logic in the underpinnings of Open World models, examining the importance of structural features of such as non-commutativity, degrees of similarity, induction, and the impact of observation. Quantum mechanics is not a problem around the edges of classical theory, but is rather a secure bridgehead in the world of science to come

Quantum and Classical Combinatorial Optimizations Applied to Lattice-Based Factorization

The availability of working quantum computers has led to several proposals and claims of quantum advantage. In 2023, this has included claims that quantum computers can successfully factor large integers, by optimizing the search for nearby integers whose prime factors are all small. This paper demonstrates that the hope of factoring numbers of commercial significance using these methods is unfounded. Mathematically, this is because the density of smooth numbers (numbers all of whose prime factors are small) decays exponentially as n grows. Our experimental reproductions and analysis show that lattice-based factoring does not scale successfully to larger numbers, that the proposed quantum enhancements do not alter this conclusion, and that other simpler classical optimization heuristics perform much better for lattice-based factoring. However, many topics in this area have interesting applications and mathematical challenges, independently of factoring itself. We consider particular cases of the CVP, and opportunities for applying quantum techniques to other parts of the factorization pipeline, including the solution of linear equations modulo 2. Though the goal of factoring 1000-bit numbers is still out-of-reach, the combinatoric landscape is promising, and warrants further research with more circumspect objectives

Quantum Circuit Components for Cognitive Decision-Making

This paper demonstrates that some nonclassical models of human decision-making can be run successfully as circuits on quantum computers. Since the 1960s, many observed cognitive behaviors have been shown to violate rules based on classical probability and set theory. For example, the order in which questions are posed affects whether participants answer 'yes' or 'no', so the population that answers `yes' to both questions cannot be modeled as the intersection of two fixed sets. It can however be modeled as a sequence of projections carried out in different orders. This and other examples have been described successfully using quantum probability, which relies on comparing angles between subspaces rather than volumes between subsets. Now in the early 2020s, quantum computers have reached the point where some of these quantum cognitive models can be implemented and investigated on quantum hardware, representing the mental states in qubit registers, and the cognitive operations and decisions using different gates and measurements. This paper develops such quantum circuit representations for quantum cognitive models, focusing particularly on modeling order effects and decision-making under uncertainty. The claim is not that the human brain uses qubits and quantum circuits explicitly (just like the use of Boolean set theory does not require the brain to be using classical bits), but that the mathematics shared between quantum cognition and quantum computing motivates the exploration of quantum computers for cognition modelling. Key quantum properties include superposition, entanglement, and collapse, as these mathematical elements provide a common language between cognitive models, quantum hardware, and circuit implementations

Embedding Probabilities in Predication Space with Hermitian Holographic Reduced Representations

Abstract. Predication-based Semantic Indexing (PSI) is an approach to generating high-dimensional vector representations of concept-relation-concept triplets. In this paper, we develop a variant of PSI that accommodates estimation of the probability of encountering a particular predication (such as fluoxetine TREATS major depressive disorder) in a collection of predications concerning a concept of interest (such as major depressive disorder). PSI leverages reversible vector transformations provided by representational approaches known as Vector Symbolic Architectures (VSA). To embed probabilities we develop a novel VSA variant, Hermitian Holographic Reduced Representations, with improvements in predictive modeling experiments. The probabilistic interpretation this facilitates reveals previously unrecognized connections between PSI and quantum theory -perhaps most notably that PSI's estimation of relatedness across multiple reasoning pathways corresponds to the estimation of the probability of traversing indistinguishable pathways in accordance with the rules of quantum probability

Cartels via the modiclus

Rosenmüller J, Sudhölter P. Cartels via the modiclus. Working Papers. Institute of Mathematical Economics. Vol 320. Bielefeld: Center for Mathematical Economics; 2000

Real, complex, and binary semantic vectors

This paper presents a combined structure for using real, complex, and binary valued vectors for semantic representation. The theory, implementation, and application of this structure are all significant. For the theory underlying quantum interaction, it is important to develop a core set of mathematical operators that describe systems of information, just as core mathematical operators in quantum mechanics are used to describe the behavior of physical systems. The system described in this paper enables us to compare more traditional quantum mechanical models (which use complex state vectors), alongside more generalized quantum models that use real and binary vectors. The implementation of such a system presents fundamental computational challenges. For large and sometimes sparse datasets, the demands on time and space are different for real, complex, and binary vectors. To accommodate these demands, the Semantic Vectors package has been carefully adapted and can now switch between different number types comparatively seamlessly. This paper describes the key abstract operations in our semantic vector models, and describes the implementations for real, complex, and binary vectors. We also discuss some of the key questions that arise in the field of quantum interaction and informatics, explaining how the wide availability of modelling options for different number fields will help to investigate some of these questions

Unsupervised Methods for Developing Taxonomies by Combining Syntactic and Statistical Information

This paper describes an unsupervised algorithm for placing unknown words into a taxonomy and evaluates its accuracy on a large and varied sample of words. The algorithm works by first using a large corpus to find semantic neighbors of the unknown word, which we accomplish by combining latent semantic analysis with part-of-speech information. We then place the unknown word in the part of the taxonomy where these neighbors are most concentrated, using a class-labelling algorithm developed especially for this task. This method is used to reconstruct parts of the existing WordNet database, obtaining results for common nouns, proper nouns and verbs. We evaluate the contribution made by part-of-speech tagging and show that automatic filtering using the class-labelling algorithm gives a fourfold improvement in accuracy