lnu.sePublications
Change search
Link to record
Permanent link

Direct link
BETA
Nyman, Peter
Publications (10 of 15) Show all publications
Nyman, P. (2011). On relations between classical and quantum theories of information and probability. (Doctoral dissertation). Växjö, Kalmar: Linnaeus University Press
Open this publication in new window or tab >>On relations between classical and quantum theories of information and probability
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Växjö, Kalmar: Linnaeus University Press, 2011. p. 161
Series
Linnaeus University Dissertations ; 60/2011
Keywords
Born’s rule, Clifford algebra, Deutsch-Josza algorithm, Grover’s algorithm, Hyperbolic interferences, Inverse Born’s rule problem, Probabilistic data, Quantum computing, Quantum error-correcting, Quantum-like representation algorithm, Shor’s algorithm, Simon’s algorithm, Simulation of quantum algorithms
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-13830 (URN)978-91-86491-98-7 (ISBN)
Public defence
2011-09-22, Weber, Universitetsplatsen 1, Växjö, 14:15 (English)
Opponent
Supervisors
Available from: 2011-08-18 Created: 2011-08-18 Last updated: 2011-08-18Bibliographically approved
Nyman, P. (2011). On the consistency of the quantum-like representation algorithm for hyperbolic interference.. Advances in Applied Clifford Algebras, 21(4), 799-811
Open this publication in new window or tab >>On the consistency of the quantum-like representation algorithm for hyperbolic interference.
2011 (English)In: Advances in Applied Clifford Algebras, ISSN 0188-7009, Vol. 21, no 4, p. 799-811Article in journal (Refereed) Published
Abstract [en]

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).

Keywords
Born’s rule problem, hyperbolic interference, hyper trigonometric interference, inverse order of conditioning, quantum-like representation algorithm.
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-13820 (URN)10.1007/s00006-011-0287-3 (DOI)2-s2.0-80455173631 (Scopus ID)
Note

Online First™, 16 March 2011

Available from: 2011-08-18 Created: 2011-08-18 Last updated: 2019-05-23Bibliographically approved
Nyman, P. & Basieva, I. (2011). Quantum-like representation algorithm for trichotomous observables.. International journal of theoretical physics, 50(12), 3864-3881
Open this publication in new window or tab >>Quantum-like representation algorithm for trichotomous observables.
2011 (English)In: International journal of theoretical physics, ISSN 0020-7748, E-ISSN 1572-9575, Vol. 50, no 12, p. 3864-3881Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
Born’s rule, Probabilistic data, Quantum-like, Trichotomous observables
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-13823 (URN)10.1007/s10773-011-0934-4 (DOI)2-s2.0-81155130798 (Scopus ID)
Available from: 2011-08-18 Created: 2011-08-18 Last updated: 2018-05-16Bibliographically approved
Nyman, P. (2010). On consistency of the quantum-like representation algorithm.. International journal of theoretical physics, 49(1), 1-9
Open this publication in new window or tab >>On consistency of the quantum-like representation algorithm.
2010 (English)In: International journal of theoretical physics, ISSN 0020-7748, E-ISSN 1572-9575, Vol. 49, no 1, p. 1-9Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
Quantum-like representation algorithm, Inverse Born's rule problem, Order of conditioning, Unitary equivalence of representations
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-13816 (URN)10.1007/s10773-009-0171-2 (DOI)
Available from: 2011-08-18 Created: 2011-08-18 Last updated: 2017-12-08Bibliographically approved
Nyman, P. & Basieva, I. (2010). Representation of probabilistic data by complex probability amplitudes: the case of triple-valued observables.. Paper presented at International Conference Advances in Quantum Theory Location: Vaxjo, SWEDEN Date: JUN 14-17, 2010. AIP Conference Proceedings, 1327, 439-449
Open this publication in new window or tab >>Representation of probabilistic data by complex probability amplitudes: the case of triple-valued observables.
2010 (English)In: AIP Conference Proceedings, ISSN 0094-243X, E-ISSN 1551-7616, Vol. 1327, p. 439-449Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
American Institute of Physics (AIP), 2010
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-13821 (URN)978-0-7354-0882-1 (ISBN)
Conference
International Conference Advances in Quantum Theory Location: Vaxjo, SWEDEN Date: JUN 14-17, 2010
Available from: 2011-08-18 Created: 2011-08-18 Last updated: 2018-05-16Bibliographically approved
Nyman, P. (2009). A Compact Code for Simulations of Quantum Error Correction in Classical Computers. Paper presented at FOUNDATIONS OF PROBABILITY AND PHYSICS—5, Växjö (Sweden),24–27 August 2008. AIP Conference Proceedings, 1101, 355-358
Open this publication in new window or tab >>A Compact Code for Simulations of Quantum Error Correction in Classical Computers
2009 (English)In: AIP Conference Proceedings, ISSN 0094-243X, E-ISSN 1551-7616, Vol. 1101, p. 355-358p. 355-358Article in journal (Other academic) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Melvill, New York: American Institute of Physics (AIP), 2009. p. 355-358
Keywords
Quantum algorithms, error corrections, quantum computers, simulation
National Category
Other Mathematics
Research subject
Natural Science, Physics; Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-16736 (URN)10.1063/1.3109961 (DOI)978-0-7354-0636-0 (ISBN)
Conference
FOUNDATIONS OF PROBABILITY AND PHYSICS—5, Växjö (Sweden),24–27 August 2008
Available from: 2012-01-12 Created: 2012-01-12 Last updated: 2017-12-08Bibliographically approved
Nyman, P. (2009). A compact program code for simulations of quantum algorithms in classical computers.. In: Igor V. Yevseyev (Ed.), Laser Physics. Paper presented at the Seventeenth International Laser Physics Workshop (LPHYS’08).
Open this publication in new window or tab >>A compact program code for simulations of quantum algorithms in classical computers.
2009 (English)In: Laser Physics / [ed] Igor V. Yevseyev, 2009Conference paper, Published paper (Other academic)
Abstract [en]

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.

National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-13827 (URN)
Conference
the Seventeenth International Laser Physics Workshop (LPHYS’08)
Available from: 2011-08-18 Created: 2011-08-18 Last updated: 2011-08-24Bibliographically approved
Nyman, P. (2009). A Symbolic Classical Computer Language for Simulation of Quantum Algorithms. Paper presented at 3rd International Symposium on Quantum Interaction, German Res Ctr Artificial Intelligence, Saarbrucken, GERMANY, MAR 25-27, 2009. Lecture Notes in Computer Science, 5494, 158-173
Open this publication in new window or tab >>A Symbolic Classical Computer Language for Simulation of Quantum Algorithms
2009 (English)In: Lecture Notes in Computer Science, ISSN 0302-9743, E-ISSN 1611-3349, Vol. 5494, p. 15p. 158-173Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Berlin / Heidelberg: Springer, 2009. p. 15
Keywords
Deutsch-Josza algorithm, Grover’s algorithm, Quantum computing, Quantum error-correcting, Shor’s algorithm, Simon’s algorithm, Simulation of quantum algorithms
National Category
Other Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:vxu:diva-6218 (URN)10.1007/978-3-642-00834-4_14 (DOI)978-3-642-00833-7 (ISBN)
Conference
3rd International Symposium on Quantum Interaction, German Res Ctr Artificial Intelligence, Saarbrucken, GERMANY, MAR 25-27, 2009
Available from: 2009-11-09 Created: 2009-11-09 Last updated: 2017-12-12Bibliographically approved
Nyman, P. (2008). Representation of Quantum Algorithms with Symbolic Language and Simulation on Classical Computer. (Licentiate dissertation). School of Mathematics and System Engineering, Växjö University
Open this publication in new window or tab >>Representation of Quantum Algorithms with Symbolic Language and Simulation on Classical Computer
2008 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [sv]

Utvecklandet av kvantdatorn är ett ytterst lovande projekt som kombinerar teoretisk och experimental kvantfysik, matematik, teori om kvantinformation och datalogi. Under första steget i utvecklandet av kvantdatorn låg huvudintresset på att skapa några algoritmer med framtida tillämpningar, klargöra grundläggande frågor och utveckla en experimentell teknologi för en leksakskvantdator som verkar på några kvantbitar. Då dominerade förväntningarna om snabba framsteg bland kvantforskare. Men det verkar som om dessa stora förväntningar inte har besannats helt. Många grundläggande och tekniska problem som dekoherens hos kvantbitarna och instabilitet i kvantstrukturen skapar redan vid ett litet antal register tvivel om en snabb utveckling av kvantdatorer som verkligen fungerar. Trots detta kan man inte förneka att stora framsteg gjorts inom kvantteknologin. Det råder givetvis ett stort gap mellan skapandet av en leksakskvantdator med 10-15 kvantregister och att t.ex. tillgodose de tekniska förutsättningarna för det projekt på 100 kvantregister som aviserades för några år sen i USA. Det är också uppenbart att svårigheterna ökar ickelinjärt med ökningen av antalet register. Därför är simulering av kvantdatorer i klassiska datorer en viktig del av kvantdatorprojektet. Självklart kan man inte förvänta sig att en kvantalgoritm skall lösa ett NP-problem i polynomisk tid i en klassisk dator. Detta är heller inte syftet med klassisk simulering. Den klassiska simuleringen av kvantdatorer kommer att täcka en del av gapet mellan den teoretiskt matematiska formuleringen av kvantmekaniken och ett förverkligande av en kvantdator. Ett av de viktigaste problemen i vetenskapen om kvantdatorn är att utveckla ett nytt symboliskt språk för kvantdatorerna och att anpassa redan existerande symboliska språk för klassiska datorer till kvantalgoritmer. Denna avhandling ägnas åt en anpassning av det symboliska språket Mathematica till kända kvantalgoritmer och motsvarande simulering i klassiska datorer. Konkret kommer vi att representera Simons algoritm, Deutsch-Joszas algoritm, Grovers algoritm, Shors algoritm och kvantfelrättande koder i det symboliska språket Mathematica. Vi använder samma stomme i alla dessa algoritmer. Denna stomme representerar de karaktäristiska egenskaperna i det symboliska språkets framställning av kvantdatorn och det är enkelt att inkludera denna stomme i framtida algoritmer.

Abstract [en]

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.

Place, publisher, year, edition, pages
School of Mathematics and System Engineering, Växjö University, 2008. p. 78
Series
Reports from MSI, ISSN 1650-2647 ; 08092
Keywords
Deutsch-Josza algorithm, Grover's algorithm, Quantum computing, Quantum error-correcting, Shor's algorithm, Simon's algorithm, Simulation of quantum algorithms, Deutsch-Josza algoritm, Grovers algoritm, Kvantdatorer, kvantmekanisk felrättande kod, Shors algoritm, Simons algoritm, Simulering av kvantdatorer
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:vxu:diva-2329 (URN)
Presentation
2008-09-29, D1136, 14:00 (English)
Opponent
Supervisors
Available from: 2008-10-06 Created: 2008-10-06 Last updated: 2017-09-01Bibliographically approved
Nyman, P. (2008). Simulation of Simon’s Algorithm in Mathematica. Quantum Computers and Computing, 8(1), 126-133
Open this publication in new window or tab >>Simulation of Simon’s Algorithm in Mathematica
2008 (English)In: Quantum Computers and Computing, ISSN 1607-9817, Vol. 8, no 1, p. 126-133Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Moscow: , 2008
Keywords
Mathematica, Simon’s Algorithm, Quantum Algorithm Simulation, Quantum Computing
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:vxu:diva-6221 (URN)
Available from: 2009-11-09 Created: 2009-11-09 Last updated: 2013-07-29Bibliographically approved
Organisations

Search in DiVA

Show all publications