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Area of Science:

  • Geometry
  • Theoretical Physics
  • Quantum Computing

Background:

  • Systems differing at short scales can share macroscopic behavior (universality classes).
  • Complexity geometry uses Riemannian geometry to study quantum computational complexity.

Purpose of the Study:

  • Classify homogeneous metrics on group manifolds by long-distance properties.
  • Investigate universality in quantum complexity definitions.

Main Methods:

  • Analyzing metrics on low- and high-dimensional Lie groups.
  • Applying concepts of geometric universality to quantum complexity.

Main Results:

  • Many Lie group metrics with different short-distance properties exhibit similar long-distance behavior.
  • Evidence suggests this phenomenon is robust, especially in higher dimensions.
  • A large universality class of quantum complexity definitions is proposed, linearly related.

Conclusions:

  • A new effective metric may emerge in complexity geometry, independent of microscopic details.
  • Findings have implications for quantum gravity conjectures and understanding quantum computational complexity.