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  3. Anticoncentration And State Design Of Random Tensor Networks.
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Anticoncentration and State Design of Random Tensor Networks.

Guglielmo Lami1, Jacopo De Nardis1, Xhek Turkeshi2

  • 1Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, CY Cergy Paris Université, 95302 Cergy-Pontoise Cedex, France.

Physical Review Letters
|February 6, 2025

View abstract on PubMed

Summary
This summary is machine-generated.

Quantum random tensor networks exhibit Haar-random behavior when bond dimensions scale polynomially with system size. This applies to both one and two-dimensional systems, showing convergence to unitary designs.

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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.