Towards complexity of primary-deformed Virasoro circuits

TL;DR

This paper investigates the complexity of Virasoro circuits in conformal field theories using the Fubini-Study metric, providing universal expressions for their time evolution and linking geometric structures to circuit complexity.

Contribution

It introduces a universal formula for the circuit complexity of primary-deformed Virasoro circuits and connects the Fubini-Study metric to Virasoro coadjoint orbits.

Findings

Complexity saturates for time-independent sources.

Large inhomogeneities increase circuit cost.

Fubini-Study metric relates to Virasoro coadjoint orbits.

Abstract

The Fubini-Study metric is a central element of information geometry. We explore the role played by information geometry for determining the circuit complexity of Virasoro circuits and their deformations. To this effect, we study unitary quantum circuits generated by the Virasoro algebra and Fourier modes of a primary operator. Such primary-deformed Virasoro circuits can be realized in two-dimensional conformal field theories, where they provide models of inhomogeneous global quenches. We consider a cost function induced by the Fubini-Study metric and provide a universal expression for its time-evolution to quadratic order in the primary deformation for general source profiles. For circuits generated by the Virasoro zero mode and a primary, we obtain a non-zero cost only if spatial inhomogeneities are sufficiently large. In this case, we find that the cost saturates when the source becomes time-independent. The exact saturation value is determined by the history of the source profile. As a byproduct, returning to undeformed circuits, we relate the Fubini-Study metric to the K\"ahler metric on a coadjoint orbit of the Virasoro group.