Universality in long-distance geometry and quantum complexity

In physics, two systems that radically differ at short scales can exhibit strikingly similar macroscopic behaviour: they are part of the same long-distance universality class(1). Here we apply this viewpoint to geometry and initiate a program of classifying homogeneous metrics on group manifolds(2) by their long-distance properties. We show that many metrics on low-dimensional Lie groups have markedly different short-distance properties but nearly identical distance functions at long distances, and provide evidence that this phenomenon is even more robust in high dimensions. An application of these ideas of particular interest to physics and computer science is complexity geometry(3-7)-the study of quantum computational complexity using Riemannian geometry. We argue for the existence of a large universality class of definitions of quantum complexity, each linearly related to the other, a much finer-grained equivalence than typically considered. We conjecture that a new effective metric emerges at larger complexities that describes a broad class of complexity geometries, insensitive to various choices of microscopic penalty factors. We discuss the implications for recent conjectures in quantum gravity. Many different homogeneous metrics on Lie groups, which may have markedly different short-distance properties, are shown to exhibit nearly identical distance functions at long distances, suggesting a large universality class of definitions of quantum complexity.