Classification on the Computational Complexity of Spin Models

TL;DR

This paper offers a comprehensive classification of the computational complexity of classical spin models, highlighting their limitations and implications for physics and quantum computation.

Contribution

It provides a unifying framework for classifying spin models based on system parameters and discusses the impact of NP-complete models on physics and computational methods.

Findings

Full classification of spin models by complexity

Implications of NP-complete models in physics

Limitations of numerical methods due to complexity

Abstract

In this note, we provide a unifying framework to investigate the computational complexity of classical spin models and give the full classification on spin models in terms of system dimensions, randomness, external magnetic fields and types of spin coupling. We further discuss about the implications of NP-complete Hamiltonian models in physics and the fundamental limitations of all numerical methods imposed by such models. We conclude by a brief discussion on the picture when quantum computation and quantum complexity theory are included.