Toward a Definition of Complexity for Quantum Field Theory States

Area of Science:

• Quantum Information Science
• Condensed Matter Physics
• Quantum Field Theory

Background:

• Understanding quantum state complexity is crucial for quantum information processing and fundamental physics.
• Gaussian states are fundamental in quantum field theories and tensor network methods like the multiscale entanglement renormalization ansatz (MERA).
• Existing complexity measures often struggle with continuous systems.

Purpose of the Study:

• To propose and investigate a novel method for quantifying the complexity of Gaussian states in continuous many-body quantum systems.
• To explore the mathematical structure of quantum state complexity using the Fubini-Study metric.
• To compare complexity measures in continuous quantum field theories with holographic complexity proposals.

Main Methods:

• Utilized the Fubini-Study metric to define a complexity measure for quantum states.
• Focused on Gaussian states, including ground states of free quantum field theories.
• Minimized complexity with respect to momentum-preserving quadratic generators forming su(1,1) algebras.
• Analyzed the geometric properties of the Gaussian state manifold under these operations.

Main Results:

• The Fubini-Study metric on the manifold of Gaussian states factorizes into hyperbolic planes.
• Minimal complexity circuits correspond to geodesics on these hyperbolic planes.
• The proposed complexity measure shows striking similarities to holographic complexity proposals.
• The findings hold for quantum field theories beyond the regime of known Einstein-gravity duals.

Conclusions:

• The Fubini-Study metric provides a viable and insightful approach to quantifying complexity in continuous quantum systems.
• The identified similarities suggest deeper connections between quantum information complexity and spacetime geometry.
• This work offers a new perspective on quantum state complexity applicable to various quantum many-body systems.