ARTICLE | doi:10.20944/preprints202203.0183.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Fermat’s last theorem; Hilbert arithmetic; Kochen and Specker’s theorem; Peano arithmetic; quantum information; qubit Hilbert space
Online: 14 March 2022 (10:54:07 CET)
In a previous paper (https://dx.doi.org/10.2139/ssrn.3648127 ), an elementary and thoroughly arithmetical proof of Fermat’s last theorem by induction has been demonstrated if the case for “n = 3” is granted as proved only arithmetically (which is a fact a long time ago), furthermore in a way accessible to Fermat himself though without being absolutely and precisely correct. The present paper elucidates the contemporary mathematical background, from which an inductive proof of FLT can be inferred since its proof for the case for “n = 3” has been known for a long time. It needs “Hilbert mathematics”, which is inherently complete unlike the usual “Gödel mathematics”, and based on “Hilbert arithmetic” to generalize Peano arithmetic in a way to unify it with the qubit Hilbert space of quantum information. An “epoché to infinity” (similar to Husserl’s “epoché to reality”) is necessary to map Hilbert arithmetic into Peano arithmetic in order to be relevant to Fermat’s age. Furthermore, the two linked semigroups originating from addition and multiplication and from the Peano axioms in the final analysis can be postulated algebraically as independent of each other in a “Hamilton” modification of arithmetic supposedly equivalent to Peano arithmetic. The inductive proof of FLT can be deduced absolutely precisely in that Hamilton arithmetic and the pransfered as a corollary in the standard Peano arithmetic furthermore in a way accessible in Fermat’s epoch and thus, to himself in principle. A future, second part of the paper is outlined, getting directed to an eventual proof of the case “n=3” based on the qubit Hilbert space and the Kochen-Specker theorem inferable from it.
ARTICLE | doi:10.20944/preprints202112.0507.v1
Subject: Mathematics & Computer Science, Logic Keywords: equality; Lewis Carroll’s paradox; Liar’s paradox; paradox of the arrow; “Achilles and the Turtle”; Hilbert arithmetic; qubit Hilbert space
Online: 31 December 2021 (11:02:32 CET)
Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll’s paradox) and that of the arrow (for “Achilles and the Turtle”), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure of both, which can be called “ontological”, on which basis “motion” studied by physics and “conclusion” studied by logic can be unified being able to bridge logic and physics philosophically in a Hegelian manner: even more, the bridge can be continued to mathematics in virtue of (2), which forces the equality (for its property of transitivity) of any two quantities to be postponed analogically ad lib and ad infinitum. The paper shows that Hilbert arithmetic underlies naturally Lewis Carroll’s paradox admitting at least three interpretations linked to each other by it: mathematical, physical and logical. Thus, it can be considered as both generalization and solution of his paradox therefore naturally unifying the completeness of quantum mechanics (i.e. the absence of hidden variables) and eventual completeness of mathematics as the same and isomorphic to the completeness of propositional logic in relation to set theory as a first-order logic (in the sense of Gödel (1930)’s completeness theorems).
ARTICLE | doi:10.20944/preprints202106.0570.v1
Subject: Arts & Humanities, Anthropology & Ethnography Keywords: bit and qubit, classical and quantum information, epoché, physical and mathematical transcendentalism, qubit space (qubit Hilbert space), separable complex Hilbert space, the totality, transcendental time
Online: 23 June 2021 (11:12:11 CEST)
Information can be considered as the most fundamental, philosophical, physical and mathematical concept originating from the totality by means of physical and mathematical transcendentalism (the counterpart of philosophical transcendentalism). Classical and quantum information, particularly by their units, bit and qubit, correspond and unify the finite and infinite. As classical information is relevant to finite series and sets, as quantum information, to infinite ones. A fundamental joint relativity of the finite and infinite, of the external and internal is to be investigated. The corresponding invariance is able to define physical action and its quantity only on the basis of information and especially: on the relativity of classical and quantum information. The concept of transcendental time, an epoché in relation to the direction of time arrow can be defined. Its correlate is that information invariant to the finite and infinite, therefore unifying both classical and quantum information.
ARTICLE | doi:10.20944/preprints202209.0358.v1
Subject: Mathematics & Computer Science, Information Technology & Data Management Keywords: Quantum Search; Qubit Management; Iterative Search
Online: 23 September 2022 (05:25:28 CEST)
Recent advances in quantum computing systems attract tremendous attention. Commercial companies, such as IBM, Amazon, and IonQ, have started to provide access to noisy intermediate-scale quantum computers. Researchers and entrepreneurs attempt to deploy their applications that aim to achieve a quantum speedup. Grover’s algorithm and quantum phase estimation are the foundations of many applications with the potential for such a speedup. While these algorithms, in theory, obtain marvelous performance, deploying them on existing quantum devices is a challenging task. For example, quantum phase estimation requires extra qubits and a large number of controlled operations, which are impractical due to low-qubit and noisy hardware. To fully utilize the limited onboard qubits, we develop a distributed application with a key-value data structure based on Grover’s algorithm called IQuCS . Consider a database with duplicates. By encoding each element to a binary type with a unique key and forming a key-value pair, we can count the number of occurrences of each element in the database based on quantum computing. We have optimized the operation process by filtering data points to make it more efficient. To determine the effect of this optimization, we evaluate it with datasets of different sizes and with different numbers of duplicates. With the assistance of classical computers, IQuCS can reduce the problem set for each query. Due to this reduction, IQuCS requires fewer qubits. Through the iterative management, IQuCS achieves a reduction of qubit virtualized consumption, up to 66.2%, with reasonable accuracy.
ARTICLE | doi:10.20944/preprints202109.0006.v1
Subject: Mathematics & Computer Science, Information Technology & Data Management Keywords: semantics; process cycle; subjectivity; quantum cognition; qubit
Online: 1 September 2021 (11:24:31 CEST)
The paper describes a model of subjective goal-oriented semantics extending standard "view-from-nowhere" approach. Generalization is achieved by using a spherical vector structure essentially supplementing the classical bit with circular dimension, organizing contexts according to their subjective causal ordering. This structure, known in quantum theory as qubit, is shown to be universal representation of contextual-situated meaning at the core of human cognition. Subjective semantic dimension, inferred from fundamental oscillation dynamics, is discretized to six process-stage prototypes expressed in common language. Predicted process-semantic map of natural language terms is confirmed by the open-source word2vec data.
ARTICLE | doi:10.20944/preprints202111.0379.v1
Subject: Behavioral Sciences, Cognitive & Experimental Psychology Keywords: core affect; emotion; semantics; process cycle; quantum cognition; qubit
Online: 22 November 2021 (11:04:58 CET)
The paper describes model of human affect based on quantum theory of semantics. The model considers emotion as subjective representation of behavioral context relative to a basis binary choice, organized by cyclical process structure and an orthogonal evaluation axis. The resulting spherical space, generalizing well-known circumplex models, accommodates basic emotions in specific angular domains. Predicted process-semantic structure of affect is observed in the word2vec data, as well as in the previously obtained spaces of emotion concepts. The established quantum-theoretic structure of affective space connects emotion science with quantum models of cognition and behavior, opening perspective for synergetic progress in these fields.
REVIEW | doi:10.20944/preprints201807.0129.v1
Subject: Materials Science, Nanotechnology Keywords: gallium nitride; rare earth ions; europium; photoluminescence; photochromism; qubit
Online: 9 July 2018 (11:05:42 CEST)
Europium is the most-studied and least-well-understood rare earth ion (REI) dopant in GaN. While attempting to increase the efficiency of red GaN light-emitting diodes (LEDs) by implanting Eu+ into p-type GaN templates, the Strathclyde University group, in collaboration with IST Lisbon and Unipress Warsaw, discovered hysteretic photochromic switching (HPS) in the photoluminescence spectrum of doubly doped GaN(Mg):Eu. Our recent work, summarised in this contribution, has used time-, temperature- and light-induced changes in the Eu intra-4f shell emission spectrum to deduce the microscopic nature of the Mg-Eu defects that form in this material. As well as shedding light on the Mg acceptor in GaN, we propose a possible role for these emission centres in quantum information and computing.
ARTICLE | doi:10.20944/preprints201703.0217.v1
Subject: Mathematics & Computer Science, General & Theoretical Computer Science Keywords: entanglement; qubit; qutrit; concurrence; negativity; relative entropy of entanglement
Online: 30 March 2017 (11:47:17 CEST)
Quantum Computers are provisioned as very high performance computers using quantum mechanical aspects for information processing. Quantum Information Theory and Quantum Computing topics are popular topics in academia aiming the construction of theoretical background of Quantum Computers. The information processing units for Quantum Computers are defined as qubits but for some problems three level (trinary) systems may be applied as well. We call these three level systems as qutrits. The scope of this work is the analysis of well-defined entanglement measures Negativity and Relative Entropy of Entanglement (REE) for two qutrit (3- level/trinary) quantum systems. In this manner, for randomly generated 1000 two qutrit and 1000 two qubit states the mentioned measures are calculated and these values are compared. These comparisons are analyzed and for quantum state ordering problem, some interesting results are reported.
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Yang-Baxter equation; braid group; qubit; ternary; polyadic; braiding quantum gate
Online: 12 July 2021 (09:08:06 CEST)
A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang-Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement which can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in details. We also consider exotic noninvertible gates which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be noninvertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and $5$-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state there exist higher braiding gates which are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.
ARTICLE | doi:10.20944/preprints202002.0338.v2
Subject: Behavioral Sciences, Cognitive & Experimental Psychology Keywords: Semantics and meaning; Context representation; Quantum cognition; Subjectivity; Quantum phase; Behavioral modeling; Qubit
Online: 22 December 2020 (11:58:16 CET)
The paper describes an algorithm for semantic representation of behavioral contexts relative to a dichotomic decision alternative. The contexts are represented as quantum qubit states in two-dimensional Hilbert space visualized as points on the Bloch sphere. The azimuthal coordinate of this sphere functions as a one-dimensional semantic space in which the contexts are accommodated according to their subjective relevance to the considered uncertainty. The contexts are processed in triples defined by knowledge of a subject about a binary situational factor. The obtained triads of context representations function as stable cognitive structure at the same time allowing a subject to model probabilistically-variative behavior. The developed algorithm illustrates an approach for quantitative subjectively-semantic modeling of behavior based on conceptual and mathematical apparatus of quantum theory.
ARTICLE | doi:10.20944/preprints202104.0306.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: communication; information; tri-state, Galois field; quantum; qubit; qutrit; qudit; trust; interconnect; out-of-band
Online: 12 April 2021 (13:06:38 CEST)
Communication, compression of information, transmission of information through noisy channels, interconnecting different information systems, cryptography, gate construction –– these areas all depend on classical information theory. We show that, in classical terms, semantic aspects of communication are not at all irrelevant to the engineering problem, contrary to Shannon, and affect the message intended to be transmitted. This is revisited and captured by an analogy to trust, in that they are essential to the channel (for proper use), but cannot be transferred (under risk of flaws) through that same channel. Information is also described by, at least, a tri-state system — not by binary logic. The trust analogy semantics can be coded as the Curry-Howard relationship, connecting computer code with structural logic, by way of different categories. Two-state and Boolean logic (aka Shannon semantics) was used classically before, with Shannon theory, but without trust analogy semantics – found to be a sine qua non condition. This is now familiar in classical gate construction with physical systems with, e.g., Verilog and SystemVerilog. The applications to computation and quantum theory are further explored. The most fundamental entity in today`s theory of information is proposed to use at least three logical states, not bits, in all applications, including: cyber-physical systems, devices, in computation, and in quantum theory.
ARTICLE | doi:10.20944/preprints202207.0279.v1
Subject: Mathematics & Computer Science, General & Theoretical Computer Science Keywords: quantum cryptography; quantum physics; quantum key distribution (QKD); B92 protocol; eavesdropper detection method; qubit; photon polarization
Online: 19 July 2022 (07:10:30 CEST)
This study presents the B92 Quantum Key Distribution scheme and its optimization and implementation considerations. Simulation software was developed to model and analyze the B92 protocol under varying conditions. Another contribution of this work is investigating how this protocol runs on real quantum computing hardware. New eavesdropping schemes were proposed and modeled.