  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.
