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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
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 1, which allows a user to explore a graphical theory by constructing graphs and performing automated and semiautomated rewrites. 1