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Exploring a Quantum Theory with Graph Rewriting and Computer Algebra
"... Abstract. Graphical languages provide a powerful tool for describing the behaviour of quantum systems. While the use of graphs vastly reduces the complexity of many calculations [4,10], manual graphical manipulation quickly becomes untenable for large graphs or large numbers of graphs. To combat thi ..."
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Abstract. Graphical languages provide a powerful tool for describing the behaviour of quantum systems. While the use of graphs vastly reduces the complexity of many calculations [4,10], manual graphical manipulation quickly becomes untenable for large graphs or large numbers of graphs. To combat this issue, we are developing a tool called Quantomatic, which allows automated and semiautomated explorations of graph rewrite systems and their underlying semantics. We emphasise in this paper the features of Quantomatic that interact with a computer algebra system to discover graphical relationships via the unification of matrix equations. Since these equations can grow exponentially with the size of the graph, we use this method to discover small identities and use those identities as graph rewrites to expand the theory. 1
Graph rewrite systems for classical structures in daggersymmetric monoidal categories
, 2008
"... This paper introduces several related graph rewrite systems derived from known identities on classical structures within a †symmetric monoidal category. First, we develop a rewrite system based on a single classical structure, and use it to develop a proof of the socalled “spidertheorem, ” where ..."
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This paper introduces several related graph rewrite systems derived from known identities on classical structures within a †symmetric monoidal category. First, we develop a rewrite system based on a single classical structure, and use it to develop a proof of the socalled “spidertheorem, ” where a connected graph containing a single classical structure can be uniquely determined by the number of inputs and outputs (i.e. it can be rewritten as a graph resembling an nlegged spider). These spiders are shown to be the normal forms of graphs containing a single classical structure. Next, complementary classical structures are introduced, as well as a new rewrite system on graphs of red and green spiders. A proof of convergence is given for a limited twocolour rewrite system, as well as insights into ways to approach normalisation in a more powerful rewrite system. Acknowledgements I would first like to thank my supervisor Ross Duncan. It has been a privilege working with him for the past six months, as his knowledge and energy have been inspirational to me and pivotal in the creation of this work. To him I give my deepest thanks for the
Monoidal Categories, Graphical Reasoning, and Quantum Computation
, 2009
"... Graphs provide a natural mechanism for visualising many algebraic systems. They are particularly useful for describing algebras in a monoidal category, such as frobenius algebras and bialgebras, which play a vital role in quantum computation. In this context, terms in the algebra are represented as ..."
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Graphs provide a natural mechanism for visualising many algebraic systems. They are particularly useful for describing algebras in a monoidal category, such as frobenius algebras and bialgebras, which play a vital role in quantum computation. In this context, terms in the algebra are represented as graphs, and algebraic identities as graph rewrite rules. To describe the theory of a more powerful monoidal algebra, one needs a concise way to define infinite sets of rules. This is addressed by introducing pattern graphs and pattern graph rewriting. An algorithm for matching is described. This is implemented in a tool called Quantomatic, which allows a user to explore a graphical theory by constructing graphs and performing automated and semiautomated rewrites.