# Path integral optimization as circuit complexity

## TL;DR

This paper connects the geometric approach to complexity in quantum field theory with the path integral optimization method in 2D conformal field theories, demonstrating how Weyl rescaling relates to circuit complexity and Liouville action.

## Contribution

It explicitly links path integral optimization to gate counting complexity, providing a concrete realization within the standard framework for 2D conformal field theories.

## Key findings

- Weyl rescaling deformation relates to circuit complexity.
- Liouville action emerges as a cost function in gate counting.
- Path integral optimization can be understood through standard complexity measures.

## 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.