A diagram is introduced for visualizing matrix product states which makestransparent a connection between matrix product factorizations of states andoperators, and complex weighted finite state automata. It is then shown how onecan proceed in the opposite direction: writing an automaton that ``generates''an operator gives one an immediate matrix product factorization of it. Matrixproduct factorizations have the advantage of reducing the cost of computingexpectation values by facilitating caching of intermediate calculations. Thusour connection to complex weighted finite state automata yields insight intowhat allows for efficient caching in matrix product algorithms. Finally, thesetechniques are generalized to the case of multiple dimensions.
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