Abstract
AbstractIn physics, two systems that radically differ at short scales can exhibit strikingly similar macroscopic behaviour: they are part of the same long-distance universality class1. Here we apply this viewpoint to geometry and initiate a program of classifying homogeneous metrics on group manifolds2 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 geometry3–7—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.
Publisher
Springer Science and Business Media LLC
Reference45 articles.
1. Wilson, K. G. The renormalization group: critical phenomena and the Kondo problem. Rev. Mod. Phys. 47, 773–840 (1975).
2. Milnor, J. Curvatures of left invariant metrics on Lie groups. Adv. Math. 21, 293–329 (1976).
3. Nielsen, M. A. A geometric approach to quantum circuit lower bounds. Preprint at https://arxiv.org/abs/quant-ph/0502070 (2005).
4. Nielsen, M. A., Dowling, M., Gu, M. & Doherty, A. C. Quantum computation as geometry. Science 311, 1133–1135 (2006).
5. Nielsen, M. A., Dowling, M. R., Gu, M. & Doherty, A. C. Optimal control, geometry, and quantum computing. Phys. Rev. A 73, 062323 (2006).
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献