Anticoncentration and State Design of Random Tensor Networks

Area of Science:

• Quantum Information Theory
• Condensed Matter Physics
• Many-Body Physics

Background:

• Tensor network states are crucial for simulating quantum many-body systems.
• Understanding the properties of random tensor networks is key to their application in quantum information.
• Random matrix product states (RMPS) and projected entangled pair states (PEPS) are important classes of tensor networks.

Purpose of the Study:

• To investigate the delocalization properties of quantum random tensor network states.
• To derive analytical expressions for the inverse participation ratio (IPR) in random matrix product states (RMPS).
• To determine the convergence of random tensor networks to Haar-random behavior and unitary designs.

Main Methods:

• Derivation of an exact analytical expression for the inverse participation ratio (IPR) for RMPS.
• Analysis of overlaps probability distribution for varying bond dimensions.
• Numerical computation of the frame potential to measure distance from the Haar ensemble.
• Extension of analysis to two-dimensional systems using random projected entangled pair states (PEPS).

Main Results:

• An exact analytical expression for the IPR of RMPS was derived for open and closed boundary conditions.
• For bond dimensions χ∼γN, the overlaps probability distribution converges to the Porter-Thomas distribution as γ increases.
• Numerical evidence shows random MPS and PEPS approximate Haar-like behavior and unitary designs for χ≫sqrt[N].
• These properties hold regardless of the spatial dimension.

Conclusions:

• Random tensor networks with polynomially scaling bond dimensions are fully Haar anticoncentrated.
• These states approximate unitary designs, a significant finding for quantum information processing.
• The study provides a comprehensive understanding of the statistical properties of random tensor networks.