Black holes, complexity and quantum chaos

TL;DR

This work extends Nielsen’s geometric framework for quantum complexity to finite-temperature and conformal settings, introducing penalty-augmented metrics on the unitary manifold to study complexity growth in black holes and chaotic systems. It identifies two growth modes—simple (symmetry-driven) and operator growth— and shows symmetry flows yield tractable costs, while chaos and Lyapunov dynamics govern the exponential growth of complexity near horizons. In SYK, a concrete lower bound links operator growth to chaos, with long-time behavior transitioning to linear growth after scrambling in agreement with gravitational backreaction and Lloyd’s bound. The results propose a bulk–boundary correspondence where growth of energy and average scaling dimensions track each other, and they establish a framework for comparing gravity duals via complexity growth patterns. Overall, the paper provides a quantifiable bridge between geometric complexity, quantum chaos, and holographic duality, anchored by symmetry analysis and SYK insights.

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.