Embracing divergence: a formalism for when your semiring is simply not complete, with applications in quantum simulation

Abstract

There is a fundamental difficulty in generalizing weighted automata to the case of infinite words: in general the infinite sum-of-products from which the weight of a given word is derived will diverge. Many solutions to this problem have been proposed, including restricting the type of weights used and employing a different valuation function that forces convergence. In this paper we describe an alternative approach that, rather than seeking to avoid the inevitable divergences, instead embraces them as a source of useful information. Specifically, rather than taking coefficients from an arbitrary semiring S we instead take them from S^N. Doing this is useful because it gives us information about how the weight of an infinite word does or does not diverge, and if it does diverge what form the divergence takes --- e.g., polynomial, exponential, etc. This approach has proved to be incredibly useful in the field of quantum simulation because when studying infinite systems, information about how quantities of interest, such as energy or magnetization, diverge is exactly what we want. In this paper we introduce a new kind of automaton which we call a diverging automaton that maps infinite words to sequences of weights from a semiring and which employs a Buchi-like boundary condition. We then develop a theory for diverging power series and prove a Kleene Theorem connecting rational diverging power series to diverging automata. Afterward we repeat this process by introducing bidiverging automata which map biinfinite words to elements in S^(Z x N), developing a theory for bidiverging power series, and proving another Kleene Theorem. We conclude by describing how bidiverging automata are applied to simulate biinfinite quantum systems.