Computational Complexity and Black Hole Horizons

TL;DR

This paper posits a duality between gravitational geometry near black hole horizons and quantum computational complexity, linking horizon distance to complexity and interpreting the ER=EPR bridge as a conduit governed by scrambling dynamics. It models black-hole microstates as fast-scrambling qubit circuits, derives how precursor complexity grows and signals propagate through the ERB, and uses the layered stretched horizon to connect spatial distance with the complexity per qubit. The analysis shows that creating firewalls requires extremely high, finely-tuned complexity, making such events unlikely for black holes formed by collapse or for typical two-sided AdS setups, though recurrences and evaporating dynamics can alter this picture. Overall, the work highlights a deep interplay between gravity, entanglement, and computational complexity, offering a framework to study interior reconstruction and sub-planckian physics with potential broader implications for holography and quantum chaos.

Abstract

Computational complexity is essential to understanding the properties of black hole horizons. The problem of Alice creating a firewall behind the horizon of Bob's black hole is a problem of computational complexity. In general we find that while creating firewalls is possible, it is extremely difficult and probably impossible for black holes that form in sudden collapse, and then evaporate. On the other hand if the radiation is bottled up then after an exponentially long period of time firewalls may be common. It is possible that gravity will provide tools to study problems of complexity; especially the range of complexity between scrambling and exponential complexity.

Figures (18)

• Figure 1: 1) A system is run forward from a state $i$ for a time $t.$ The sign of the Hamiltonian is flipped and the system evolves for another time $t.$ At the end the final state $f$ is the same as the initial state. The symbol $I$ indicates that nothing was done at the end of the first interval to disturb the configuration. 2) The operator $W$ is inserted before the evolution is reversed.
• Figure 2: Penrose diagram showing the layered structure of the near-horizon region of an ADS black hole.
• Figure 3: A degree of freedom at the outer boundary of the stretched horizon may be represented in terms of boundary degrees of freedom along the vertical base of the yellow triangle.
• Figure 4: A degree of freedom at the Planck distance from the horizon is represented in terms of boundary degrees of freedom between the red lines.
• Figure 5: As time evolves symmetrically in the eternal black hole, the ERB is stretched in length. The stretching takes place near the horizons.