Complexity and shock wave geometries

In this paper we refine a conjecture relating the time-dependent size of an Einstein-Rosen bridge (ERB) to the computational complexity of the dual quantum state. Our refinement states that the complexity is proportional to the spatial volume of the ERB. More precisely, up to an ambiguous numerical coefficient, we propose that the complexity is the regularized volume of the largest codimension one surface crossing the bridge, divided by $G_N{l}_{\mathrm{AdS}}$. We test this conjecture against a wide variety of spherically symmetric shock wave geometries in different dimensions. We find detailed agreement.

Figure 1

The yellow region is the Wheeler–De Witt patch for the times $t_L,t_R$. The brown curve indicates a spacelike surface connecting the two boundaries.

Figure 2

Maximum volume surface for infinite $t_{L,R}$.

Figure 3

Qubit model for the TFD state. The TFD consists of a product of Bell pairs shared between the left and right sides.

Figure 4

In the left panel the operation $U^{†}(t)IU$ is illustrated. The letters $i$ and $f$ represent initial and final states. The backtracking trajectories illustrate the cancellation of $U$ and $U^{†}$. In the right panel the unit insertion is replaced by the insertion of $W$. The backtracking of trajectories takes place for a limited time until the butterfly effect kicks in at the scrambling time.

Figure 5

Evolution of the ERB tensor network. The red curves depict the RT surface for computing vertical entanglement. The tensor network fills the volume of the ERB.

Figure 6

Time fold with six insertions. The insertions at $t_1,t_2,t_3,t_4$ and $t_6$ occur at switchback points. The insertion at $t_5$ does not.

Figure 7

Kruskal diagram for an ERB dual to the TFD state perturbed by out-of-time-order operators at the left boundary. The blue curve represents the maximal-volume surface crossing the ERB. Each point of intersection with a shock has two sets of $(u,v)$ coordinates: one in the patch to the left, and one in the patch to the right. These are related by null shifts determined by the strength of each shock.

Figure 8

Folded time axis and geometry for two shocks with $t_1<t_2<t_L$. The $t_2$ insertion is through going, and the associated shock runs right beside the stronger $t_1$ shock.

Figure 9

For a maximally entangled state there is an equivalence between acting with unitary operators on the left and right. The two shock wave geometries shown in the figure would be equivalent.

Figure 10

The dark blue curves are the maximal surfaces defining $V(t_L,t_R)$, $V(t_L,v_R)$, and $V(u_L,v_R)$. The pale blue curve is the limiting infinite-time maximal surface.