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RESEARCH PROJECT
QUANTUM COMPUTATION:
PARALLELISM AND VISUALISATION
(082-0982562-3160)
Head: Mladen Pavicic
DESCRIPTION OF RESEARCH
9.0 RESEARCH PLAN, PROCEDURES
AND METHODS
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9.2 |
Research procedures, protocol and plan (10 000 characters): |
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In this section we shall refer to the four parts of the project defined at the beginning of Sec. 9.1 by invoking their numbers: (1)-(4). (1) There are still only a few quantum algorithms whose complexity does not grow exponentially with the number or size of variables. All of them are based on the Fourier transform and therefore that they all can be reduced to finding eigenvalues and eigenvectors of unitary operators: 10.5.7. Such eigenvalue problems amount to solving Schrodinger equations. However, it also shows that it is most promising to start with algorithms that would calculate genuine quantum problems and that a search for "classical" and universal Fourier transform algorithms might greatly benefit from any new algorithm of the former kind. Hence, we will focus on algorithms that would calculate and simulate quantum systems and enable quantum parallel computing. We shall work on algorithms for obtaining Fourier transformation for excited atom states used in quantum computation experiments. To this aim we start with our recent ground breaking algorithm in the field of algebra and graph theory that is also applicable to quantum systems in the Hilbert space and classical ones in the phase space. The procedure is as follows. Each Hilbert space vector can be described by means of vertex in a graph. Relations between vectors, e.g., orthogonality, can be described be edges that connect vertices. Hence, we ascribe graphs to nonlinear equations which describe relations between vectors. To each unknown it corresponds a vertex and to each equation an edge. Vectors in quantum Hilbert or classical phase space satisfy some properties, e.g., they are in a particular state or are ascribed some values. These state and values we can measure in the end. The novelty of our approach is that these state and values can be defined on (linear) graph if it allows this. We then obtain an algorithm for exhaustive selection of those graphs that allows chosen states or values. We discard all the remaining graphs. In this way we substitute the graph elimination for solving equations. Remaining graphs (extremely small number of them survives) correspond to nonlinear equations that can have solutions. Then we search for solutions among them using new algorithms based on linear interval analysis. The role of each co-worker from the project in its particular parts is given in (2) and (3) below. We will use results obtained in 10.5.2,7,9,10,12. New algorithms and programs will be developed for obtaining a visualisation of Fourier transform algorithm outputs since the characteristic distribution of peaks in the outputs of Fourier transform algorithm turn out to be highly significant for their handling and application. (2) Quantum gates are well described by simple Hilbert space formalisms and all gates can eventually be reduced to two-level ones: "Two-level gates are universal." But algorithms we use to implement quantum computation by means of quantum gates assume discrete inputs in and outputs from a quantum computer. So a problem emerges with a description of quantum systems and processes by means of finite dimensional algebras so as to enable a direct introduction of the description into a quantum computer. For, continuous quantum states (e.g., position of an electron, radial distribution of an electron cloud, etc.) imply a representation by means of an infinite-dimensional Hilbert space and it cannot be introduced directly into a discrete quantum computer with discrete quantum gates - we mean qubits - even when they are superposed and entangled. With such quantum algebras at hand, the difference between a classical problem introduced into a classical computer and a quantum problem introduced into a quantum computer by means of the algebras would be the following. Any classical problem we want to calculate we first have to digitalize, i.e., express in Boolean algebra (formulate with the help of 0,1 strings) and only then introduce into a classical computer. A quantum problem, on the other hand, originally and genuinely expressed by means of a quantum algebra could be introduced directly into a quantum computer whose language is quantum algebra in the same sense in which Boolean algebra is the language of a classical computer. Implementation of quantum problems into a continuous quantum computer requires a different approach. [2] The main tool for finding new algebras will be the numerous algorithms and programs we developed for generating and evaluating quantum algebras, Hilbert lattices, Greechie diagrams, MMP diagrams, and Kochen-Specker states we developed over the last 15 years. In particular we will use varieties of program nauty developed by our foreign coworker at the project, Brendan McKay, Professor of Computer Science at the Department of Computer Science at the Australian National University and programs lattice, oml, beran, and greechie developed by our foreign coworker Norman D. Megill. To find new quantum algebras we have to carry out massive parallel calculations on clusters and the grid as we already did in coming to our previous algebras. [26] We will also develop a visualization of a graphical representation of algebras in analogy to the ones we used in Refs. [26,27] The way to achieve implementation of quantum problems into a continuous quantum computer is via novel quantization of the quasi-classical description of particle-field interaction that we also developed. [2] (3) The ability of quantum computers to calculate exponentially faster than today's classical computers depends on the efficiency of the quantum error-correction scheme that can correct the states faster than they can decohere. Our algorithms for exhaustive generation of arbitrary Kochen-Specker systems can be re-elaborated so as to generate arbitrary error-correction schemes. It stems from our recognition of a correspondence between error correction codes - both classical and quantum - and geometry of graphs. To achieve this goal we will use 1. Interval analysis programs and the library ALIAS for solving nonlinear equations, developed by our foreign co-worker at the project, Jean-Pierre Merlet, Head of the COPRIN project at the INRIA, the French national institute for research in computer science and control 2. Other varieties of program nauty developed by Brendan McKay (see above) 3. Programs states and states01 developed by Norman D. Megill (see above). 4. Numerical methods for solving systems of nonlinear algebraic equations resulting from the unit partition method and a method of discrete elements used in discontinuous deformation analysis elaborated by our co-worker Kresimir Fresl in new discrete civil engineering problems: 10.5.4,5,27,28. They boil down to numerical methods of solving nonlinear systems. The methods will be incorporated into our graph approach to solving nonlinear equations. 5. Programs for visualization of the codes we are going to develop. (4) To test the obtained algebras and error-correction schemes we will theoretically design fundamental experiments with a particular emphasis on engineering and controlling atom states and their superposition and the properties of photons interaction with photons. In particular the time windows required for controlling states and interaction will give us estimates of decoherence time and required level of error correction. We plan to analyse (a) atom-photon coupling, (b) superposition of atom states, (c) atom-photon and atom-atom entanglements, (d) quantum gates that use (a)-(c). The experiments will partly rely on our previous the ground breaking results: Pavicic and Summhammer, PRL, 73, 3191 (1994); Pavicic, J. Opt. Soc. Am. B 12, 821 (1995), Pavicic, Phys. Lett. A 223, 241 (1996) that were carried out by different experimental groups in their labs in the meantime. A good example for such a theoretical elaboration is 10.5.1. Parts (1)-(3) of the project require massive parallel calculations on clusters and the grid. The project should be a part a bigger program Distributed Processing and Scientific Data Visualization at the Institute Rudjer Boskovic in Zagreb. The project and program should (through the University Computing Centre) join the EGEE-II project (Enabling Grids for E-sciencE). Thus the present project will be one of the nodes of the program. The University Computing Centre will contribute to networking issues in EGEE-II project. We also expect to be included in the European GRID. |
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10.0 PRESENT STATE, CONTRIBUTION
AND COMPETENCY OF
RESEARCHER
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10.1 |
Previous discoveries (2 500 characters): |
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(1) Quantum algorithms (2) Quantum algebras and theoretical description of quantum systems (3) Quantum error correction (4) Quantum experiments are (1) Practically all known quantum algorithms are based on Fourier transforms: 10.5.7. Therefore D. S. Abrams and S. Lloyd, Phys. Rev. Lett. 79, 2586, (1997) and C. Zalka, Proc. Roy. Soc. London A, 454, 313 (1998) designed first algorithms for finding eigenvalues and eigenvectors of simple unitary operators under particular restrictions. They have shown the algorithms provide an exponential speed increase compared to the standard way of solving such problems on a classical computer. We will generalize their approach to elaborate on Fourier transforms algorithms for atom states used in quantum computation experiments. (2) No additional universal quantum algebra is needed for quantum gates themselves: 10.5.7. However, such algebras would enable us to implement any Schrodinger equation directly into a quantum computer. To obtain such an algebra will take a Hilbert lattice approach. [G. Kalmbach, Measures and Hilbert Lattices, Singapore, World Scientific, 1986] We rely on new classes of algebras directly corresponding to quantum states and properties of Hilbert spaces: 10.5.13,24. (3) If we encode a single qubit in the state of superposition by means of other entangled qubits whose sequence corresponds to classical codewords, we will be able to use classical error correction applied to a superposition of such quantum codewords. [A. Steane, PRL, 77, 793 (1996), A. R. Calderbank and P. W. Shor, PRA, 54, 1098 (1996)] We will use our previous discovery of an algorithm for exhaustive generation of Kochen-Specker vectors, 10.5.9,10,12,29, to obtain algorithms for generating quantum error-correction codes. Our final goal is obtain an algorithm for exhaustive generation of maximally efficient error-correction codes. (4) We will use the previous discoveries of photon entanglements [C. H. Bennett, Phys. Rev. Lett. 70, 1895 (1993); M. Zukowski at al., Phys. Rev. Lett. 71, 4287 (1993); Pavicic, M., Phys. Rev. A 50, 3486 (1994); Pavicic, M. and J. Summhammer, Phys. Rev. Lett. 73, 3191 (1994); Pavicic, M., J. Opt. Soc. Am. B 12, 821 (1995)]. We will also use previous discoveries of the resonator interaction-free measurement and its control of atomic states and superposition: Pavicic, M., Phys. Lett. A 223, 241 (1996); Paul, H. and Pavicic, M., J. Opt. Soc. Am. B 14, 1275 (1997), 10.5.1 (2007). |
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