Path integral optimization as circuit complexity

TL;DR

This work bridges path-integral optimization and circuit complexity in two-dimensional CFTs by showing that Weyl deformations of the Euclidean path integral can be realized within the standard gate-counting framework. The Liouville action $S_L$ emerges as the $ abla$-dependent cost for the Weyl factor $\omega$, obtained from a controlled expansion of a suitably chosen cost functional, thereby linking path-integral optimization to circuit complexity in a concrete way. It provides a precise mapping from curved-background path integrals to circuits generated by the energy-momentum components $h$ and $p$ and discusses the interpretation of a DBI-like action in a special foliation. The results clarify the role of covariance and UV regularization, and point toward future directions for fully covariant cost measures, higher-order corrections, and extensions to holography and mixed states.

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.

Figures (2)

• Figure 1: Euclidean strip over which we perform the path integral \ref{['eq:A']}. The left image represents the flat geometry \ref{['eq:flat']}, while the right represents the Weyl-rescaled background \ref{['eq.Weylresc']}. Note that we hold the boundary conditions at $\tau=0$ and $\tau=\beta$ fixed. This guarantees that we produce the same operator $\rho_{\beta}$, up to an overall normalization governed by the exponent of the Liouville action \ref{['eq:SL']}.
• Figure 2: $\Sigma_\tau$ and $\Sigma_t$ denote time slices of the background geometry in $(\tau,x)$ coordinates, see eq. \ref{['eq.Weylresc']}, and $(t,y)$ coordinates, see eq. \ref{['eq:gMV']}, respectively. At a given point, the Euclidean angle between tangents to $\Sigma_{t}$ and $\Sigma_{\tau}$ is given by the function $\sigma(\tau,x)$, as defined in eq. \ref{['eq.defsigma']}. On the right side of the figure, $a$ and $b$ appear as defined by eq. \ref{['eq:gMV']}, which illustrates their geometrical interpretation.