Complexity continuum within Ising formulation of NP problems

TL;DR

This paper investigates the complexity spectrum of Ising model instances used in NP-hard optimization problems, identifying criteria for easy versus hard instances and exploring their implications for physical Ising machines.

Contribution

It introduces an optimisation simplicity criterion to distinguish computationally easy instances within NP-hard Ising problems and analyzes their complexity using circulant coupling matrices.

Findings

Easy instances can be identified using the proposed criterion.

Optical and photonic systems can efficiently solve simple instances.

Rewiring matrices can increase problem complexity.

Abstract

A promising approach to achieve computational supremacy over the classical von Neumann architecture explores classical and quantum hardware as Ising machines. The minimisation of the Ising Hamiltonian is known to be NP-hard problem for certain interaction matrix classes, yet not all problem instances are equivalently hard to optimise. We propose to identify computationally simple instances with an `optimisation simplicity criterion'. Such optimisation simplicity can be found for a wide range of models from spin glasses to k-regular maximum cut problems. Many optical, photonic, and electronic systems are neuromorphic architectures that can naturally operate to optimise problems satisfying this criterion and, therefore, such problems are often chosen to illustrate the computational advantages of new Ising machines. We further probe an intermediate complexity for sparse and dense models by analysing circulant coupling matrices, that can be `rewired' to introduce greater complexity. A compelling approach for distinguishing easy and hard instances within the same NP-hard class of problems can be a starting point in developing a standardised procedure for the performance evaluation of emerging physical simulators and physics-inspired algorithms.