This study focuses on students' way of reasoning about a proof in mathematics. The experiences of teaching students in the beginning of their studies at universities show that students have an obstacle in using deductive methods. The students' activity was designed specifically to investigate their deductive ability and to see if they can develop their way of reasoning. The group activities and interviews follow the students from the beginning where they, with great enthusiasm, begin colouring maps as a first sketch to a complete proof. The well-known statement to prove is chosen from a field in mathematics that the students are unfamiliar with, namely graph theory. More precisely it concerns the number of possible colourings of maps. Some university students have problems with constructing proofs, but in many cases the teacher can help them to reach a deductive reasoning.
This study considers implementations of error correction in a simulation language on a classical computer. Error correction will be necessarily in quantum computing and quantum information. We will give some examples of the implementations of some error correction codes.These implementations will be made in a more general quantum simulation language on a classical computer in the language Mathematica. The intention of this research is to develop a programming language that is able to make simulations of all quantum algorithms and error corrections in the same framework. The program code implemented on a classical computer will provide a connection between the mathematical formulation of quantum mechanics and computational methods. This gives us a clear uncomplicated language for the implementations of algorithms.
A general quantum simulation language on a classical computer provides the opportunity to compare an experiential result from the development of quantum computers with mathematical theory. The intention of this research is to develop a program language that is able to make simulations of all quantum algorithms in same framework. This study examines the simulation of quantum algorithms on a classical computer with a symbolic programming language. We use the language Mathematica to make simulations of well-known quantum algorithms. The program code implemented on a classical computer will be a straight connection between the mathematical formulation of quantum mechanics and computational methods. This gives us an uncomplicated and clear language for the implementations of algorithms. The computational language includes essential formulations such as quantum state, superposition and quantum operator. This symbolic programming language provides a universal framework for examining the existing as well as future quantum algorithms. This study contributes with an implementation of a quantum algorithm in a program code where the substance is applicable in other simulations of quantum algorithms.
Quantum computing is an extremely promising research combining theoretical and experimental quantum physics, mathematics, quantum information theory and computer science. Classical simulation of quantum computations will cover part of the gap between the theoretical mathematical formulation of quantum mechanics and the realization of quantum computers. One of the most important problems in “quantum computer science” is the development of new symbolic languages for quantum computing and the adaptation of existing symbolic languages for classical computing to quantum algorithms. The present paper is devoted to the adaptation of the Mathematica symbolic language to known quantum algorithms and corresponding simulation on the classical computer. Concretely we shall represent in the Mathematica symbolic language Simon’s algorithm, the Deutsch-Josza algorithm, Grover’s algorithm, Shor’s algorithm and quantum error-correcting codes. We shall see that the same framework can be used for all these algorithms. This framework will contain the characteristic property of the symbolic language representation of quantum computing and it will be a straightforward matter to include this framework in future algorithms.
In this paper we continue to study so-called “inverse Born’s rule problem”: to constructa representation of probabilistic data of any origin by a complex probability amplitudewhich matches Born’s rule. The corresponding algorithm—quantum-like representation algorithm(QLRA)—was recently proposed by A. Khrennikov (Found. Phys. 35(10):1655–1693, 2005; Physica E 29:226–236, 2005; Dokl. Akad. Nauk 404(1):33–36, 2005; J. Math.Phys. 46(6):062111–062124, 2005; Europhys. Lett. 69(5):678–684, 2005). Formally QLRAdepends on the order of conditioning. For two observables (of any origin, e.g., physical orbiological) a and b, b|a- and a|b conditional probabilities produce two representations, sayin Hilbert spaces Hb|a and Ha|b. In this paper we prove that under “natural assumptions”(which hold, e.g., for quantum observables represented by operators with nondegeneratespectra) these two representations are unitary equivalent. This result proves the consistencyof QLRA.
The quantum–like representation algorithm (QLRA) was introduced by A. Khrennikov to solve the “inverse Born’s rule problem”, i.e. to construct a representation of probabilistic data– measured in any context of science– and represent this data by a complex or more general probability amplitude which matches a generalization of Born’s rule. The outcome from QLRA will introduce the formula of total probability with an additional term of trigonometric, hyperbolic or hyper-trigonometric interference and this is in fact a generalization of the familiar formula of interference of probabilities. We study representation of statistical data (of any origin) by a probability amplitude in a complex algebra and a Clifford algebra (algebra of hyperbolic numbers). The statistical datas are collected from measurements of two trichotomous observables and the complexity of the problem increased eventually compared to the case of dichotomous observables.We see that only special statistical data (satisfying a number of nonlinear constraints) have a quantum–like representation. In this paper we will present a class of statistical data which satisfy these nonlinear constraints and have a quantum–like representation. This quantum–like representation induces trigonometric-, hyperbolic- and hyper–trigonometric interferences representation.
In this thesis we study quantum-like representation and simulation of quantum algorithms by using classical computers.The quantum--like representation algorithm (QLRA) was introduced by A. Khrennikov (1997) to solve the ``inverse Born's rule problem'', i.e. to construct a representation of probabilistic data-- measured in any context of science-- and represent this data by a complex or more general probability amplitude which matches a generalization of Born's rule.The outcome from QLRA matches the formula of total probability with an additional trigonometric, hyperbolic or hyper-trigonometric interference term and this is in fact a generalization of the familiar formula of interference of probabilities.
We study representation of statistical data (of any origin) by a probability amplitude in a complex algebra and a Clifford algebra (algebra of hyperbolic numbers). The statistical data is collected from measurements of two dichotomous and trichotomous observables respectively. We see that only special statistical data (satisfying a number of nonlinear constraints) have a quantum--like representation.
We also study simulations of quantum computers on classical computers.Although it can not be denied that great progress have been made in quantum technologies, it is clear that there is still a huge gap between the creation of experimental quantum computers and realization of a quantum computer that can be used in applications. Therefore the simulation of quantum computations on classical computers became an important part in the attempt to cover this gap between the theoretical mathematical formulation of quantum mechanics and the realization of quantum computers. Of course, it can not be expected that quantum algorithms would help to solve NP problems for polynomial time on classical computers. However, this is not at all the aim of classical simulation.
The second part of this thesis is devoted to adaptation of the Mathematica symbolic language to known quantum algorithms and corresponding simulations on classical computers. Concretely we represent Simon's algorithm, Deutsch-Josza algorithm, Shor's algorithm, Grover's algorithm and quantum error-correcting codes in the Mathematica symbolic language. We see that the same framework can be used for all these algorithms. This framework will contain the characteristic property of the symbolic language representation of quantum computing and it will be a straightforward matter to include future algorithms in this framework.
Recently quantum-like representation algorithm (QLRA) wasintroduced by A. Khrennikov [20]–[28] to solve the so-called “inverseBorn’s rule problem”: to construct a representation of probabilistic databy a complex or hyperbolic probability amplitude or more general complextogether with hyperbolic which matches Born’s rule or its generalizations.The outcome from QLRA is coupled to the formula of totalprobability with an additional term corresponding to trigonometric, hyperbolicor hyper-trigonometric interference. The consistency of QLRAfor probabilistic data corresponding to trigonometric interference was recentlyproved [29].We complete the proof of the consistency of QLRA tocover hyperbolic interference as well. We will also discuss hyper trigonometricinterference. The problem of consistency of QLRA arises, becauseformally the output of QLRA depends on the order of conditioning. Fortwo observables (e.g., physical or biological) a and b, b|a- and a|b- conditionalprobabilities produce two representations, say in Hilbert spacesHb|a and Ha|b (in this paper over the hyperbolic algebra). We provethat under “natural assumptions” these two representations are unitaryequivalent (in the sense of hyperbolic Hilbert space).
Quantum computing is an extremely promising project combining theoretical and experimental quantum physics, mathematics, quantum information theory and computer science. At the first stage of development of quantum computing the main attention was paid to creating a few algorithms which might have applications in the future, clarifying fundamental questions and developing experimental technologies for toy quantum computers operating with a few quantum bits. At that time expectations of quick progress in the quantum computing project dominated in the quantum community. However, it seems that such high expectations were not totally justified. Numerous fundamental and technological problems such as the decoherence of quantum bits and the instability of quantum structures even with a small number of registers led to doubts about a quick development of really working quantum computers. Although it can not be denied that great progress had been made in quantum technologies, it is clear that there is still a huge gap between the creation of toy quantum computers with 10-15 quantum registers and, e.g., satisfying the technical conditions of the project of 100 quantum registers announced a few years ago in the USA. It is also evident that difficulties increase nonlinearly with an increasing number of registers. Therefore the simulation of quantum computations on classical computers became an important part of the quantum computing project. Of course, it can not be expected that quantum algorithms would help to solve NP problems for polynomial time on classical computers. However, this is not at all the aim of classical simulation. Classical simulation of quantum computations will cover part of the gap between the theoretical mathematical formulation of quantum mechanics and the realization of quantum computers. One of the most important problems in "quantum computer science" is the development of new symbolic languages for quantum computing and the adaptation of existing symbolic languages for classical computing to quantum algorithms. The present thesis is devoted to the adaptation of the Mathematica symbolic language to known quantum algorithms and corresponding simulation on the classical computer. Concretely we shall represent in the Mathematica symbolic language Simon's algorithm, the Deutsch-Josza algorithm, Grover's algorithm, Shor's algorithm and quantum error-correcting codes. We shall see that the same framework can be used for all these algorithms. This framework will contain the characteristic property of the symbolic language representation of quantum computing and it will be a straightforward matter to include this framework in future algorithms.
This study examines the simulation of Deutsch-Jozsa algorithm in Mathematica. The program code implemented on a classical computer will be a straight connection between the mathematical formulation of quantum mechanics and computational methods in Mathematica. This program code will be a foundation of a universal simulation language.
This paper is a presentation of how to implement quantum algorithms (namely, Shor's algorithm ) on a classical computer by using the well-known Mathematica package. It will give us a lucid connection between mathematical formulation of quantum mechanics and computational methods.
This study considers implementations of error correction in a simulation language on a classical computer. Error correction will be necessarily in quantum computing and quantum information. We will give some examples of the implementations of some error correction codes. These implementations will be made in a more general quantum simulation language on a classical computer in the language Mathematica.The intention of this research is to develop a programming language that is able to make simulations of all quantum algorithms and error corrections in the same framework. The program code implemented on a classical computer will provide a connection between the mathematical formulation of quantum mechanics and computational methods. This gives us a clear uncomplicated language for the implementations of algorithms.
A general quantum simulation language on a classical computer provides the opportunity to compare an experiential result from the development of quantum computers with mathematical theory. The intention of this research is to develop a program language that is able to make simulations of quantum mechanical processes as well as quantum algorithms. This study examines the simulation of quantum algorithms on a classical computer with a symbolic programming language. We use the language Mathematica to make a simulation of well-known quantum algorithms. The program code implemented on a classical computer will be a straight connection between the mathematical formulation of quantum mechanics and computational methods. This give us an uncomplicated and clear language for implementations of algorithms. The computational language includes essential formulations such as quantum state, superposition and quantum operator. This symbolic programming language provides a universal framework for examining the existing as well as future quantum algorithms. This study contributes with an implementation of a quantum algorithm in a program code where the substance is applicable in other simulation of quantum algorithms.
We study the problem of representing statistical data (of any origin) by a complex probability amplitude. This paper is devoted to representation of data collected from measurements of two trichotomous observables. The complexity of the problem eventually increases compared to the case of dichotomous observables. We see that only special statistical data (satisfying a number of nonlinear constraints) have the quantum–like representation.
The problem of inter-relation between classical and quantum probabilistic data wasdiscussed in numerous papers (from various points of view), see, e.g., [1, 2, 3, 4, 6, 5, 7,8, 14, 15]. We are interested in the problem of representation of probabilistic data of anyorigin 1 by complex probability amplitude, so to say a “wave function”. This problemwas discussed in very detail in [17]. A general QL-representation algorithm (QLRA)was presented in [17]. This algorithm is based on the formula of total probability withinterference term – a disturbance of the standard formula of total probability. Startingwith experimental probabilistic data, QLRA produces a complex probability amplitudesuch that probability can be reconstructed by using Born’s rule.Although the formal scheme of QLRA works for multi-valued observables of anarbitrary dimension, the description of the class of probabilistic data which can betransfered into QL-amplitudes (the domain of application of QLRA) depends very muchon the dimension. In [19] the simplest case of data generated by dichotomous observableswas studied. In this paper we study trichotomous observables. The complexity of theproblem increases incredibly comparing with the two dimensional case.Finally, we remark that our study is closely related to the triple slit interferenceexperiment and Sorkin’s equality [16]. This experiment provides an important test offoundations of QM.The scheme of presentation is the following one. We start with observables given byQM and derive constraints on phases which are necessary and sufficient for the QLrepresentation.Then we use these constraints to produce complex amplitudes from data(of any origin); some examples, including numerical, are given.