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Conference proceedings/Conference proceedings - IEEE International Conference on Systems, Man, and Cybernetics

Modeling Feynman's quantum computer using stochastic high level Petri nets

Leo Ojala, Elina Parviainen, Olli-Matti Penttinen, H. Beaver, Teemu Tynjälä

Published Nov 13, 2002• 2 citations

Abstract

One of the first and simplest models of a quantum computer was introduced by Richard Feynman (1985). The aim of our study is to extend the application area of Petri nets, which have been very successfully used to model classical engineering systems, by modeling the use and operation of Feynman's quantum computer. Originally, Feynman could not give a time bound for the completion of his computer's computation; a periodical measurement procedure was needed giving rise to a specific computational approach. We have chosen Feynman's NOT-computer, in the form of two /spl radic/NOT gates in series, as our object of demonstration; its use and operation are modeled in detail using stochastic high level Petri net formalism.

DOI: 10.1109/icsmc.2001.972980

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One of the first and simplest models of a quantum computer was introduced by Richard Feynman (1985). The aim of our study is to extend the application area of Petri nets, which have been very successfully used to model classical engineering systems, by modeling the use and operation of Feynman's quantum computer. Originally, Feynman could not give a time bound for the completion of his computer's computation; a periodical measurement procedure was needed giving rise to a specific computational approach. We have chosen Feynman's NOT-computer, in the form of two /spl radic/NOT gates in series, as our object of demonstration; its use and operation are modeled in detail using stochastic high level Petri net formalism.

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Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, there is a widely used and successful theory of "chemical reaction networks", which describes the interactions of molecules in a stochastic rather than quantum way. Computer science and population biology use the same ideas under a different name: "stochastic Petri nets". But if we look at these theories from the perspective of quantum theory, they turn out to involve creation and annihilation operators, coherent states and other well-known ideas - but in a context where probabilities replace amplitudes. We explain this connection as part of a detailed analogy between quantum mechanics and stochastic mechanics. We use this analogy to present new proofs of two major results in the theory of chemical reaction networks: the deficiency zero theorem and the Anderson-Craciun-Kurtz theorem. We also study the overlap of quantum mechanics and stochastic mechanics, which involves Hamiltonians that can generate either unitary or stochastic time evolution. These Hamiltonians are called "Dirichlet forms", and they arise naturally from electrical circuits made only of resistors.