Universality in long-distance geometry and quantum complexity

Abstract

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. Here we apply this viewpoint to geometry and initiate a program of classifying homogeneous metrics on group manifolds 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--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.

arXiv commentv2: journal version. new title (previously "Effective Geometry, Complexity, and Universality")

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Cited by in corpus (7)

• Quantum complexity in gravity, quantum field theory, and quantum information science
• Saturation and recurrence of quantum complexity in random local quantum dynamics
• Polynomial Equivalence of Complexity Geometries
• Universal Early-Time Growth in Quantum Circuit Complexity
• The Complexity of Being Entangled
• Quantum circuit complexity and unsupervised machine learning of topological order
• Wavefunction branches demand a definition!