Quantum Complexity and Negative Curvature

TL;DR

This paper demonstrates that classical geodesics on negatively curved surfaces exhibit complexity growth patterns similar to those in quantum systems, revealing a universal behavior across classical and quantum dynamics.

Contribution

It establishes a surprising parallel between quantum complexity growth and classical geodesic behavior on negatively curved geometries.

Findings

Classical geodesics show complexity growth similar to quantum systems.

The pattern persists under external perturbations.

Universal behavior observed across classical and quantum dynamics.

Abstract

As time passes, once simple quantum states tend to become more complex. For strongly coupled k-local Hamiltonians, this growth of computational complexity has been conjectured to follow a distinctive and universal pattern. In this paper we show that the same pattern is exhibited by a much simpler system: classical geodesics on a compact two-dimensional geometry of uniform negative curvature. This striking parallel persists whether the system is allowed to evolve naturally or is perturbed from the outside.