From Complexity Geometry to Holographic Spacetime

TL;DR

The paper establishes a principled link between a boundary complexity measure based on the square of the Fubini-Study distance $F_{FS}$ and a bulk geometric observable $F_{bulk}$ in $AdS_3$, derived from geodesic-length data via kinematic space. It shows that, for circuits generated by 2D conformal transformations, the FS cost maps to a UV-finite bulk functional valid for vacuum, conical defects, and BTZ black holes, and that this dual reproduces linear growth of complexity for time-evolved thermofield-double states. In shockwave backgrounds without bulk matter, $F_{bulk}$ captures the switchback effect, while $F_{FS}$ itself cannot; in matter-filled geometries the FS dual ceases to be exact, highlighting limits of the correspondence and pointing to rich structure in holographic complexity. The work thus provides a concrete first-principles bridge between boundary complexity geometry and holographic spacetime, with clear avenues to extend to higher dimensions, additional operator sources, JT gravity, and ensemble approaches, potentially deepening connections to entanglement structure and tensor-network pictures.

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.

Figures (6)

• Figure 1: Illustration of the map between a distance measure in the complexity geometry on the left and a geometric object in the asymptotically AdS spacetime on the right. Any two states $\ket{\psi(t_1)},\ket{\psi(t_2)}$ can be put on a Bloch sphere spanned by $\ket{\psi(t_1)}$ and an orthogonal state $\ket{\chi}$ obtained by subtracting from $\ket{\psi(t_2)}$ the part parallel to $\ket{\psi(t_1)}$, i.e. $\ket{\psi(t_2)} \propto \ket{\chi} + \langle \psi(t_1) | \psi(t_2) \rangle\ket{\psi(t_1)}$. The geodesic distance for the Fubini-Study metric between $\ket{\psi(t_1)}$ and $\ket{\psi(t_2)}$ is then the angle $\theta$ on this Bloch sphere. These two states live on different time slices at the boundary of the same AdS geometry shown on the right-hand side. The infinitesimal distance in the complexity geometry on the left between the states $\ket{\psi(t_1=t)}$ and $\ket{\psi(t_2=t+dt)}$ manifests itself as a geometric object in the AdS space on the right. Therefore, the total cost also acquires a geometric dual localized in between the two time slices $t=0$ and $t=t_f$ in the AdS space. For optimal circuits, this geometric object becomes a gravity dual to the complexity.
• Figure 2: Boundary anchored geodesics making up the kinematic space for the AdS$_3$ geometry dual to a conformally transformed vacuum state on the left and the ${\mathbb{Z}}_3$ conical defect dual to an excited state on the right.
• Figure 3: Time evolution of computational complexity as conjectured by Susskind:2014rva.
• Figure 4: Left and center: the bulk cost function $F_\text{bulk}$ plotted over $t_W$ in the localized shockwave geometry with compact horizon on a linear resp. logarithmic scale. Right: the corresponding total cost $F_\text{bulk,tot}$ (the $t_W$ integral of $F_\text{bulk}$). The plots match perfectly with the expectation from the infection models that the exponential increase, which is clearly visible in the log-scale plot in the center, turns into a linear increase at the scrambling time. All plots are evaluated for $t_*=0.5$ and $\beta = 1/4$.
• Figure 5: Left: bulk region swept out by geodesics contributing to $F_\text{bulk}$ (defined in \ref{['eq:definition-F_bulk']}). Right: union of all the bulk regions on the left for $0 \leq t \leq t_0$ contributing to the computational complexity $C_\text{FS}(t_0)$.