Path integral optimization as circuit complexity

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.

v2: added appendix on kappa=2 norm; minor changes to match PRL layout

References in corpus (9)

• Towards a derivation of holographic entanglement entropy
• A class of quantum many-body states that can be efficiently simulated
• Quantum Computation as Geometry
• Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT
• Deriving covariant holographic entanglement
• Einstein's Equations from Varying Complexity
• From Path Integrals to Tensor Networks for AdS/CFT
• Casting Shadows on Holographic Reconstruction
• A class of non-equilibrium states and the black hole interior