Path Integral Optimization as Circuit Complexity.

Hugo A Camargo,Johannes Knaute,Michal P Heller,Ro Jefferson

doi:10.1103/physrevlett.123.011601

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 prepare 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 by 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.

Highlights

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 by a concrete realization within the standard gate counting framework
Introduction.—One of the most interesting developments of the past decade is the confluence of ideas from quantum gravity, quantum field theory (QFT), and quantum information science
The impact of this research trend is perhaps most striking in holography (AdS=conformal field theories (CFTs)) [1,2,3] where, e.g., the entanglement entropy of spatial subregions in the boundary field theory has been shown to play a prominent role in describing higher-dimensional geometries; see Ref. [4] for a review

Summary

Introduction

Introduction.—One of the most interesting developments of the past decade is the confluence of ideas from quantum gravity, quantum field theory (QFT), and quantum information science. 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. The aim of the present Letter is to explicitly demonstrate that path integral optimization in its best-explored setting of two-dimensional conformal field theories (CFTs) may be given a precise formulation in terms of circuit complexity.

Objectives

Results

Conclusion