From Complexity Geometry to Holographic Spacetime

TL;DR

This paper establishes a first-principles connection between quantum complexity in conformal field theories and geometric objects in holographic AdS spacetime, enhancing understanding of the complexity-geometry duality.

Contribution

It introduces a novel duality mapping between complexity measures based on Fubini-Study distance and geometric objects in AdS spacetime, applicable to black hole geometries.

Findings

Identifies a geometric object in AdS dual to Fubini-Study based complexity.

Provides insights into linear complexity growth in black hole spacetimes.

Explains the switchback effect through the new duality.

Abstract

An important conjecture within the AdS/CFT correspondence relates holographic spacetime to the quantum computational complexity of the dual quantum field theory. However, the quantitative understanding of this relation is still an open question. In this work, we introduce and study a map between a computational complexity measure and its holographic counterpart from first principles. We consider quantum circuits built out of conformal transformations in two-dimensional conformal field theory and a complexity measure based on assigning a cost to quantum gates via the Fubini-Study distance. We find a novel geometric object in three-dimensional anti-de Sitter spacetimes that is dual to this distance. This duality also provides a more general map between holographic geometry of anti-de Sitter universes and complexity geometry as defined in information theory, in which each point represents a state and distances between states are measured by the Fubini-Study metric. We apply the newly found duality to the eternal black hole spacetime and discuss both the origin of linear growth of complexity and the switchback effect within our approach.