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Complexity for Conformal Field Theories in General Dimensions.

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We explore quantum circuit complexity for conformal field theory states. Our method connects circuit distances to geodesic lengths in anti-de Sitter space, offering new insights into quantum information geometry.

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Area of Science:

  • Theoretical Physics
  • Quantum Information Science
  • High Energy Physics

Background:

  • Conformal field theories (CFTs) are fundamental in quantum physics.
  • Quantum circuit complexity measures the effort to prepare quantum states.
  • Understanding complexity is crucial for quantum gravity and quantum information.

Purpose of the Study:

  • To develop a framework for studying quantum circuit complexity in CFTs.
  • To connect circuit complexity to geometric properties of symmetry groups.
  • To explore the relationship between quantum states and spacetime geometry.

Main Methods:

  • Utilizing unitary representations of the Lorentzian conformal group.
  • Analyzing the geometry of coadjoint orbits for distance functions.
  • Relating quantum circuits to timelike geodesics in anti-de Sitter (AdS) space.
  • Generalizing coherent states for other symmetry groups.

Main Results:

  • A novel method for calculating circuit complexity in CFTs across arbitrary dimensions.
  • Circuit complexity is shown to be equivalent to distances between timelike geodesics in AdS.
  • The geometric interpretation of distance functions is clarified through coadjoint orbit geometry.
  • The framework is extended to other symmetry groups.

Conclusions:

  • This work provides a geometric interpretation of quantum circuit complexity in CFTs.
  • The AdS/CFT correspondence provides a powerful lens for understanding quantum complexity.
  • The generalized method offers a versatile tool for studying quantum systems with various symmetries.