Complexity and Shock Wave Geometries

TL;DR

The authors refine and test a conjecture that the quantum computational complexity of the dual CFT state is proportional to the spatial volume of the Einstein-Rosen bridge, specifically C(t_L,t_R) ∝ V(t_L,t_R). They develop a precise framework for maximal-volume ERB surfaces, derive their infinite- and finite-time behavior, and apply this to a broad class of spherically symmetric shock-wave geometries across dimensions. Their analysis shows detailed agreement with the conjecture, including a scrambling-time–related cancellation pattern and a reflection principle tied to maximal entanglement, while connecting these results to tensor-network pictures. The work strengthens the link between chaos, complexity growth, and interior black-hole geometry, with implications for holography and the TN viewpoint on spacetime emergence.

Abstract

In this paper we refine a conjecture relating the time-dependent size of an Einstein-Rosen bridge to the computational complexity of the 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 $G_N l_{AdS}$. We test this conjecture against a wide variety of spherically symmetric shock wave geometries in different dimensions. We find detailed agreement.

Figures (10)

• Figure 1: The yellow region is the Wheeler-DeWitt patch for the times $t_L, t_R.$ The brown curve indicates a space-like 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 $U^\dag(t) I U$ is illustrated. The letters $i$ and $f$ represent initial and final states. The backtracking trajectories illustrate the cancelation of $U$ and $U^{\dag}.$ 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.