Black holes, complexity and quantum chaos

TL;DR

This paper explores the relationship between black holes, quantum chaos, and computational complexity by extending geometric approaches to quantify costs in quantum systems, revealing growth patterns linked to chaos and symmetry.

Contribution

It introduces an extended Nielsen geometric framework for finite temperature scenarios and analyzes complexity growth in black holes and SYK models, connecting chaos and energy scales.

Findings

Infalling particle costs relate to the maximal Lyapunov exponent.

Complexity growth in SYK saturates at long times, matching gravity predictions.

Different growth types (operator vs simple) are characterized and analyzed.

Abstract

We study aspects of black holes and quantum chaos through the behavior of computational costs, which are distance notions in the manifold of unitaries of the theory. To this end, we enlarge Nielsen geometric approach to quantum computation and provide metrics for finite temperature/energy scenarios and CFT's. From the framework, it is clear that costs can grow in two different ways: operator vs `simple' growths. The first type mixes operators associated to different penalties, while the second does not. Important examples of simple growths are those related to symmetry transformations, and we describe the costs of rotations, translations, and boosts. For black holes, this analysis shows how infalling particle costs are controlled by the maximal Lyapunov exponent, and motivates a further bound on the growth of chaos. The analysis also suggests a correspondence between proper energies in the bulk and average `local' scaling dimensions in the boundary. Finally, we describe these complexity features from a dual perspective. Using recent results on SYK we compute a lower bound to the computational cost growth in SYK at infinite temperature. At intermediate times it is controlled by the Lyapunov exponent, while at long times it saturates to a linear growth, as expected from the gravity description.