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Area of Science:

  • Quantum Many-Body Physics
  • Computational Quantum Mechanics
  • Theoretical Physics

Background:

  • Matrix product states (MPS) and projected entangled pair states (PEPS) are key tools for analyzing quantum many-body systems.
  • While MPS are well-understood, fundamental analytical and numerical challenges persist for PEPS, particularly in symmetry encoding and numerical stabilization.
  • Canonical forms are crucial for addressing these PEPS challenges.

Purpose of the Study:

  • To investigate the decidability of fundamental problems concerning projected entangled pair states (PEPS).
  • To explore the limitations of PEPS in fully understanding quantum many-body systems.
  • To determine if key issues like general symmetry encoding and numerical method stabilization are algorithmically resolvable.

Main Methods:

  • Algorithmic undecidability proofs.
  • Theoretical analysis of PEPS properties.
  • Investigation of symmetry encoding and numerical stabilization techniques.

Main Results:

  • Key problems related to PEPS, including general symmetry encoding and stabilization via canonical forms, are algorithmically undecidable.
  • These findings demonstrate that a complete and unbiased understanding of quantum many-body systems using PEPS is fundamentally limited.
  • The study exposes inherent algorithmic constraints in the application of PEPS.

Conclusions:

  • The inherent undecidability of core PEPS problems imposes fundamental limitations on the comprehensive analysis of quantum many-body systems.
  • This research highlights the boundaries of what can be systematically achieved with PEPS, impacting both analytical and numerical approaches.
  • Future research may need to explore alternative frameworks or accept inherent limitations when using PEPS.