From complexity geometry to holographic spacetime

An important conjecture within the $\mathrm{AdS}/\mathrm{CFT}$ correspondence relates holographic spacetime to the quantum computational complexity of the dual quantum field theory. However, the quantitative understanding of this relation is largely an open question. In this work, to address this question we establish 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.

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 $|\psi (t_1)⟩,|\psi (t_2)⟩$ can be put on a Bloch sphere spanned by $|\psi (t_1)⟩$ and an orthogonal state $|\chi ⟩$ obtained by subtracting from $|\psi (t_2)⟩$ the part parallel to $|\psi (t_1)⟩$, i.e. $|\psi (t_2)⟩\propto |\chi ⟩+⟨\psi (t_1)|\psi (t_2)⟩|\psi (t_1)⟩$. The geodesic distance for the Fubini-Study metric between $|\psi (t_1)⟩$ and $|\psi (t_2)⟩$ 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 $|\psi (t_1=t)⟩$ and $|\psi (t_2=t+dt)⟩$ 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 ${\mathrm{AdS}}_3$ 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 [4].

Figure 4

Left and center: the bulk cost function $F_{\mathrm{bulk}}$ plotted over $t_W$ in the localized shock wave geometry with compact horizon on a linear resp. logarithmic scale. Right: the corresponding total cost $F_{\mathrm{bulk},\mathrm{tot}}$ (the $t_W$ integral of $F_{\mathrm{bulk}}$). 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 $F_{\mathrm{bulk}}$ [defined in (25)]. Right: union of all the bulk regions on the left for $0\le t\le t_0$ contributing to the computational complexity $C_{\mathrm{FS}}(t_0)$.

Figure 6

Left: $F_{\mathrm{bulk}}$ plotted over $t_W$ and right: the corresponding total cost $F_{\mathrm{bulk},\mathrm{tot}}$ (the $t_W$ integral of $F_{\mathrm{bulk}}$) in a BTZ geometry perturbed by a single delocalized shock wave with strength $\alpha$ from (45) and $t_{*}=0.5$, $\beta =1/4$.