Complexity for Conformal Field Theories in General Dimensions

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.