Computations on quantum computers based on non-classic probability theory

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.

Contextual decisions and beliefs and their impact upon market outcomes are at the core of research in behavioural finance. We describe some of the notable probabilistic fallacies that underpin investor behaviour and the consequent deviation of asset prices from the rational expectations equilibrium. In real financial markets, the complexity of financial products and the surrounding ambiguity calls for a more general formalization of agents belief formation than offered by the standard probability theory and dynamic models based on classical stochastic processes. The main advantage of quantum probability (QP) is that it can capture contextuality of beliefs through the notion of non-commuting prospect observables. QP has the potential to model myopia in asset return evaluation, as well as inter-asset valuation. Moreover, the interference term of the agents’ comparison state can provide a quantitative description of their vacillating ambiguity perception characterized by non-additive beliefs of agents. Some of the implications of non-classicality in beliefs for the composite market outcomes can also be modelled with the aid of QP. As a final step we also discuss the contributions of the growing body of psychological studies that reveal a true (quantum type) contextuality in human preference statistics showing that the classical probability theory is too restrictive to capture the very strong non-classical correlations between preference outcomes and beliefs.

The main thesis of the present paper is that: Quantum Probability is not a generalization of classical probability, but it is a deeper level of it. Classical random variables have an intrinsic (microscopic) non–commutative structure that generalize usual quantum theory. The study of this generalization is the core of the non–linear quantization program.

Quantum Computer: Quantum Model and Reality

Previous article Next article Addendum: On Homogeneous Controlled Markov Models with Continuous Time and Finite or Countable Number of StatesA. A. Yushkevich and E. A. FainbergA. A. Yushkevich and E. A. Fainberghttps://doi.org/10.1137/1125030PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout"Addendum: On Homogeneous Controlled Markov Models with Continuous Time and Finite or Countable Number of States." Theory of Probability & Its Applications, 25(1), p. 215 Previous article Next article FiguresRelatedReferencesCited byDetails Volume 25, Issue 1| 1980Theory of Probability & Its Applications History Submitted:14 March 1979Published online:28 July 2006 InformationCopyright © 1980 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1125030Article page range:pp. 215-215ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

We extend Ruelle’s Perron-Frobenius theorem to the case of Holder continuous functions on a topologically mixing topological Markov shift with a countable number of states. LetP(ϕ) denote the Gurevic pressure of ϕ and letLϕ be the corresponding Ruelle operator. We present a necessary and sufficient condition for the existence of a conservative measure ν and a continuous functionh such thatLϕ*ν=eP(ϕ)ν,Lϕh=eP(ϕ)h and characterize the case when ∝hdν<∞. In the case whendm=hdν is infinite, we discuss the asymptotic behaviour ofLϕk, and show how to interpretdm as an equilibrium measure. We show how the above properties reflect in the behaviour of a suitable dynamical zeta function. These resutls extend the results of [18] where the case ∝hdν<∞ was studied.

Markov Processes Application for Estimation of Parameters of Assembly Lines

Quantum probability

In the light of Quantum Physics, probabilistic technique is said to support a backbone in the domain of Quantum Information Retrieval (QIR). This well known domain is based on the Quantum Probability Theory, which is used to examine the probability of relevance of a document given a user's query. The Quantum Probability Ranking Principle (QPRP), captures dependencies between documents in absolute way via “quantum interference” by considering a single vector i.e., one-dimension subspace. The HAL semantic space can be used to capture meaning of the document of information needs by representing the document into a high dimensional semantic space. This paper presents an investigation of the Quantum Probability Ranking Principle with the concept of HAL semantic space such that which can act as a bridge the gap between Quantum Probability Theory and HAL semantic space, that can results to agreeable performance in an ad-hoc retrieval task of document ranking.

Comparing large-scale graphs based on quantum probability theory

Continuous-variable (CV) devices are a promising platform for demonstrating large-scale quantum information protocols. In this framework we define a general quantum computational model based on a CV hardware. It consists of vacuum input states, a finite set of gates-including non-Gaussian elements-and homodyne detection. We show that this model incorporates encodings sufficient for probabilistic fault-tolerant universal quantum computing. Furthermore, we show that this model can be adapted to yield sampling problems that cannot be simulated efficiently with a classical computer, unless the polynomial hierarchy collapses. This allows us to provide a simple paradigm for experiments to probe quantum advantage relying on Gaussian states, homodyne detection, and some form of non-Gaussian evolution. We finally address the recently introduced model of instantaneous quantum computing in CV, and prove that the hardness statement is robust with respect to some experimentally relevant simplifications in the definition of that model.

The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray, made a first classification of such algebras into three types. The nonrelativistic quantum theory of systems with finitely many degrees of freedom deals exclusively with type I algebras. However, for the description of further quantum systems, the other types of von Neumann algebras are indispensable. The paper reviews quantum probability theory in terms of general von Neumann algebras, stressing the similarity of the conceptual structure of classical and noncommutative probability theories and emphasizing the correspondence between the classical and quantum concepts, though also indicating the nonclassical nature of quantum probabilistic predictions. In addition, differences between the probability theories in the type I, II and III settings are explained. A brief description is given of quantum systems for which probability theory based on type I algebras is known to be insufficient. These illustrate the physical significance of the previously mentioned differences.

Inter-relation between quantum and classical probability models is one of the most fundamental problems of quantum foundations. Nowadays this problem also plays an important role in quantum technologies, in quantum cryptography and the theory of quantum random generators. In this letter, we compare the viewpoint of Richard Feynman that the behavior of quantum particles cannot be described by classical probability theory with the viewpoint that quantum–classical inter-relation is more complicated (cf, in particular, with the tomographic model of quantum mechanics developed in detail by Vladimir Man'ko). As a basic example, we consider the two-slit experiment, which played a crucial role in quantum foundational debates at the beginning of quantum mechanics (QM). In particular, its analysis led Niels Bohr to the formulation of the principle of complementarity. First, we demonstrate that in complete accordance with Feynman's viewpoint, the probabilities for the two-slit experiment have the non-Kolmogorovian structure, since they violate one of basic laws of classical probability theory, the law of total probability (the heart of the Bayesian analysis). However, then we show that these probabilities can be embedded in a natural way into the classical (Kolmogorov, 1933) probability model. To do this, one has to take into account the randomness of selection of different experimental contexts, the joint consideration of which led Feynman to a conclusion about the non-classicality of quantum probability. We compare this embedding of non-Kolmogorovian quantum probabilities into the Kolmogorov model with well-known embeddings of non-Euclidean geometries into Euclidean space (e.g., the Poincaré disk model for the Lobachvesky plane).

Two-qubit entangling gates for superconducting quantum computers

Blind quantum computation (BQC) enables a client with less quantum computational ability to delegate her quantum computation to a server with strong quantum computational power while preserving the client's privacy. Generally, many-qubit entangled states are often used to complete BQC tasks. But for a large-scale entangled state, it is difficult to be described since its Hilbert space dimension is increasing exponentially. Furthermore, the number of entangled qubits is limited in experiment of existing works. To tackle this problem, in this paper we propose a universal BQC protocol based on measurement with minor resources, where the trap technology is adopted to verify correctness of the server's measurement outcomes during computation and testing process. In our model there are two participants, a client who prepares initial single-qubit states and a server that performs universal quantum computation. The client is almost classical since she does not require any quantum computational power, quantum memory. To realize the client's universal BQC, we construct an $m\times n$ latticed state composed of six-qubit cluster states and eight-qubit cluster states, which needs less qubits than the brickwork state. Finally, we analyze and prove the blindness, correctness, universality and verifiability of our proposed BQC protocol.