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Abstract

Early efforts to understand complexity in field theory have primarily
employed a geometric approach based on the concept of circuit complexity in
quantum information theory. In a parallel vein, it has been proposed that
certain deformations of the Euclidean path integral that prepares a given
operator or state may provide an alternative definition, whose connection to
the standard notion of complexity is less apparent. In this letter, we bridge
the gap between these two proposals in two-dimensional conformal field
theories, by explicitly showing how the latter approach from path integral
optimization may be given a concrete realization within the standard gate
counting framework. In particular, we show that when the background geometry is
deformed by a Weyl rescaling, a judicious gate counting allows one to recover
the Liouville action as a particular choice within a more general class of cost
functions.