scholarly journals Obscure Qubits and Membership Amplitudes

2021 ◽  
Author(s):  
Steven Duplij ◽  
Raimund Vogl
Keyword(s):  
Hilbert Space ◽  
Quantum Computing ◽  
Density Matrix ◽  
Tensor Product ◽  
Quantum Computation ◽  
Membership Function ◽  
Quantum Probability ◽  
Quantum States ◽  
Born Rule ◽  
Quantum Amplitude

We propose a concept of quantum computing which incorporates an additional kind of uncertainty, i.e. vagueness (fuzziness), in a natural way by introducing new entities, obscure qudits (e.g. obscure qubits), which are characterized simultaneously by a quantum probability and by a membership function. To achieve this, a membership amplitude for quantum states is introduced alongside the quantum amplitude. The Born rule is used for the quantum probability only, while the membership function can be computed from the membership amplitudes according to a chosen model. Two different versions of this approach are given here: the “product” obscure qubit, where the resulting amplitude is a product of the quantum amplitude and the membership amplitude, and the “Kronecker” obscure qubit, where quantum and vagueness computations are to be performed independently (i.e. quantum computation alongside truth evaluation). The latter is called a double obscure-quantum computation. In this case, the measurement becomes mixed in the quantum and obscure amplitudes, while the density matrix is not idempotent. The obscure-quantum gates act not in the tensor product of spaces, but in the direct product of quantum Hilbert space and so called membership space which are of different natures and properties. The concept of double (obscure-quantum) entanglement is introduced, and vector and scalar concurrences are proposed, with some examples being given.

scholarly journals Enhanced quantum signature scheme using quantum amplitude amplification operators

PLoS ONE ◽  
2021 ◽  
Vol 16 (10) ◽  
pp. e0258091
Author(s):  
Basma Elias ◽  
Ahmed Younes
Keyword(s):  
Quantum Computing ◽  
Quantum Signature ◽  
Quantum States ◽  
Bell States ◽  
Signature Scheme ◽  
Signature Schemes ◽  
Diffusion Operator ◽  
Quantum Amplitude ◽  
Preparation Phase ◽  
Amplitude Amplification

Quantum signature is the use of the principles of quantum computing to establish a trusted communication between two parties. In this paper, a quantum signature scheme using amplitude amplification techniques will be proposed. To secure the signature, the proposed scheme uses a partial diffusion operator and a diffusion operator to hide/unhide certain quantum states during communication. The proposed scheme consists of three phases, preparation phase, signature phase and verification phase. To confuse the eavesdropper, the quantum states representing the signature might be hidden, not hidden or encoded in Bell states. It will be shown that the proposed scheme is more secure against eavesdropping when compared with relevant quantum signature schemes.

scholarly journals Quantum States of Hierarchic Systems

2003 ◽  
Vol 01 (02) ◽  
pp. 269-278 ◽  
Author(s):  
Mikhail V. Altaisky
Keyword(s):  
Hilbert Space ◽  
Density Matrix ◽  
Degrees Of Freedom ◽  
Quantum Systems ◽  
Quantum States ◽  
Matrix Formalism ◽  
Density Matrix Formalism ◽  
Theory Of Measurements ◽  
Novel Method ◽  
Chemical And Biological Systems

The density matrix formalism which is widely used in the theory of measurements, quantum computing, quantum description of chemical and biological systems always implies the averaging over all states of the environment. In practice this is impossible because the environment of the system is the complement of this system to the whole Universe and contains infinitely many degrees of freedom. A novel method of construction density matrix which implies the averaging only over the direct environment is proposed. The Hilbert space of state vectors for the hierarchic quantum systems is constructed.

scholarly journals The Born rule as a statistics of quantum micro-events

2020 ◽  
Vol 476 (2244) ◽  
pp. 20200282
Author(s):  
Yurii V. Brezhnev
Keyword(s):  
Hilbert Space ◽  
Quantum Theory ◽  
Vector Space ◽  
Scalar Product ◽  
Mathematical Structure ◽  
Quantum States ◽  
Born Rule ◽  
Statistical Nature ◽  
Quadratic Character ◽  
Quantum Superpositions

We deduce the Born rule from a purely statistical take on quantum theory within minimalistic math-setup. No use is required of quantum postulates. One exploits only rudimentary quantum mathematics—a linear, not Hilbert’, vector space—and empirical notion of the Statistical Length of a state. Its statistical nature comes from the lab micro-events (detector-clicks) being formalized into the C -coefficients of quantum superpositions. We also comment that not only has the use not been made of quantum axioms (scalar-product, operators, interpretations , etc.), but that the involving thereof would be, in a sense, inconsistent when deriving the rule. In point of fact, the quadratic character of the statistical length, and even not (the ‘physics’ of) Born’s formula, represents a first step in constructing the mathematical structure we name the Hilbert space of quantum states.

Quantum Computing and Quantum Communication

2018 ◽  
pp. 7715-7730
Author(s):  
Göran Pulkkis ◽  
Kaj J. Grahn
Keyword(s):  
Quantum Computing ◽  
Quantum Computation ◽  
Quantum Cryptography ◽  
Quantum Algorithms ◽  
Quantum Circuit ◽  
Quantum States ◽  
Future Perspectives ◽  
Quantum Informatics ◽  
Quantum Programming ◽  
Computing Models

This article presents state-of-the-art and future perspectives of quantum computing and communication. Timeline of relevant findings in quantum informatics, such as quantum algorithms, quantum cryptography protocols, and quantum computing models, is summarized. Mathematics of information representation with quantum states is presented. The quantum circuit and adiabatic models of quantum computation are outlined. The functionality, limitations, and security of the quantum key distribution (QKD) protocol is presented. Current implementations of quantum computers and principles of quantum programming are shortly described.

Entangled Quantum States and the Kronecker Product

2002 ◽  
Vol 57 (8) ◽  
pp. 689-691 ◽  
Author(s):  
Willi-Hans Steeb ◽  
Yorick Hardy
Keyword(s):  
Hilbert Space ◽  
Quantum Computing ◽  
Quantum Teleportation ◽  
Kronecker Product ◽  
Quantum Error Correction ◽  
Quantum States ◽  
Dense Coding ◽  
Factorization Problem ◽  
Entangled Quantum States ◽  
Quantum Error

CEntangled quantum states are an important component of quantum computing techniques such as quantum error-correction, dense coding and quantum teleportation. We determine the requirements for a state in the Hilbert space ⊗ Cnfor m, n ∈ N to be entangled and a solution to the corresponding “factorization” problem if this is not the case.We consider the implications of these criteria for computer algebra applications.

scholarly journals Quantum Computation and Measurements from an Exotic Space-Time R4

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 736
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo M. Amaral ◽  
Klee Irwin
Keyword(s):  
Hilbert Space ◽  
Quantum Computing ◽  
Fundamental Group ◽  
Quantum Computation ◽  
Quantum Measurement ◽  
Space Time ◽  
Fano Plane ◽  
Topological Quantum ◽  
Universal Quantum ◽  
Complete Quantum

The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a ‘magic’ state ψ in d-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a finite ‘contextual’ geometry. In the present work, we choose G as the fundamental group π 1 ( V ) of an exotic 4-manifold V, more precisely a ‘small exotic’ (space-time) R 4 (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean R 4 ). Our selected example, due to S. Akbulut and R. E. Gompf, has two remarkable properties: (a) it shows the occurrence of standard contextual geometries such as the Fano plane (at index 7), Mermin’s pentagram (at index 10), the two-qubit commutation picture G Q ( 2 , 2 ) (at index 15), and the combinatorial Grassmannian Gr ( 2 , 8 ) (at index 28); and (b) it allows the interpretation of MICs measurements as arising from such exotic (space-time) R 4 s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of ‘quantum gravity’.

ENTANGLED QUANTUM STATES AND A C++ IMPLEMENTATION

2000 ◽  
Vol 11 (01) ◽  
pp. 69-77 ◽  
Author(s):  
WILLI-HANS STEEB ◽  
YORICK HARDY
Keyword(s):  
Hilbert Space ◽  
Quantum Computing ◽  
Error Correction ◽  
Quantum Teleportation ◽  
Quantum Error Correction ◽  
Quantum States ◽  
Dense Coding ◽  
Factorization Problem ◽  
Entangled Quantum States ◽  
Quantum Error

Entangled quantum states are an important component of quantum computing techniques such as quantum error-correction, dense coding and quantum teleportation. We determine the requirements for a state in the Hilbert space C4 to be entangled and a solution to the corresponding "factorization" problem if this is not the case. The factorization of nonentangled states is implemented in C++.

scholarly journals Probabilities and Epistemic Operations in the Logics of Quantum Computation

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 837 ◽  
Author(s):  
Maria Dalla Chiara ◽  
Hector Freytes ◽  
Roberto Giuntini ◽  
Roberto Leporini ◽  
Giuseppe Sergioli
Keyword(s):  
Quantum Information ◽  
Quantum Computation ◽  
Transition Probability ◽  
Quantum Probability ◽  
Quantum States ◽  
Quantum Version ◽  
Logical Connectives ◽  
Quantum Computational Logics ◽  
Absolute Concept ◽  
Relative Concept

Quantum computation theory has inspired new forms of quantum logic, called quantum computational logics, where formulas are supposed to denote pieces of quantum information, while logical connectives are interpreted as special examples of quantum logical gates. The most natural semantics for these logics is a form of holistic semantics, where meanings behave in a contextual way. In this framework, the concept of quantum probability can assume different forms. We distinguish an absolute concept of probability, based on the idea of quantum truth, from a relative concept of probability (a form of transition-probability, connected with the notion of fidelity between quantum states). Quantum information has brought about some intriguing epistemic situations. A typical example is represented by teleportation-experiments. In some previous works we have studied a quantum version of the epistemic operations “to know”, “to believe”, “to understand”. In this article, we investigate another epistemic operation (which is informally used in a number of interesting quantum situations): the operation “being probabilistically informed”.

Quantum Computing and Quantum Communication

2019 ◽  
pp. 1641-1659
Author(s):  
Göran Pulkkis ◽  
Kaj J. Grahn
Keyword(s):  
Quantum Computing ◽  
Quantum Computation ◽  
Quantum Cryptography ◽  
Quantum Algorithms ◽  
Quantum Circuit ◽  
Quantum States ◽  
Future Perspectives ◽  
Quantum Informatics ◽  
Quantum Programming ◽  
Computing Models

This chapter presents state-of-the-art and future perspectives of quantum computing and communication. Timeline of relevant findings in quantum informatics, such as quantum algorithms, quantum cryptography protocols, and quantum computing models, is summarized. Mathematics of information representation with quantum states is presented. The quantum circuit and adiabatic models of quantum computation are outlined. The functionality, limitations, and security of the quantum key distribution (QKD) protocol is presented. Current implementations of quantum computers and principles of quantum programming are shortly described.

The Essence of Quantum Computing

2018 ◽  
Author(s):  
Rajendra K. Bera
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Quantum Computing ◽  
Quantum Computation ◽  
Quantum Algorithms ◽  
Underlying Mechanism ◽  
Quantum States ◽  
Quantum Computers ◽  
Efficient Computation

In Part I we laid the foundation on which quantum algorithms are built. In part II we harnessed such exotic aspects of quantum mechanics as superposition, entanglement and collapse of quantum states to show how powerful quantum algorithms can be constructed for efficient computation. In Part III (the concluding part) we discuss two aspects of quantum computation: (1) the problem of correcting errors that inevitably plague physical quantum computers during computations, by algorithmic means; and (2) a possible underlying mechanism for the collapse of the wave function during measurement.

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