Projected Entangled Pair States: Fundamental Analytical and Numerical Limitations

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.