Computational complexity continuum within Ising formulation of NP problems

Kirill P Kalinin,Natalia G Berloff

doi:10.1038/s42005-021-00792-0

Kirill P Kalinin, Natalia G Berloff

Open Access

https://doi.org/10.1038/s42005-021-00792-0

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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 yet not all problem instances are equivalently hard to optimise. Given that the operational principles of Ising machines are suited to the structure of some problems but not others, we propose to identify computationally simple instances with an ‘optimisation simplicity criterion’. Neuromorphic architectures based on optical, photonic, and electronic systems can naturally operate to optimise instances satisfying this criterion, which are therefore often chosen to illustrate the computational advantages of new Ising machines. As an example, we show that the Ising model on the Möbius ladder graph is ‘easy’ for Ising machines. By rewiring the Möbius ladder graph to random 3-regular graphs, we probe an intermediate computational complexity between P and NP-hard classes with several numerical methods. Significant fractions of polynomially simple instances are further found for a wide range of small size models from spin glasses to maximum cut problems. 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.

Highlights

A promising approach to achieve computational supremacy over the classical von Neumann architecture explores classical and quantum hardware as Ising machines
We provide numerical evidence of such optimisation simplicity for instances covering a wide range of problems from spin glass models to 3-regular maximum cut (MaxCut) problems
With the understanding of what is essential for an individual instance of the NP-hard problem to be counted as simple, we present a natural approach for restoring complexity and study the continuous complexity transition from simple to hard instances for Ising optimisation on physical Ising machines

Summary

Introduction

A promising approach to achieve computational supremacy over the classical von Neumann architecture explores classical and quantum hardware as Ising machines. Neuromorphic architectures based on optical, photonic, and electronic systems can naturally operate to optimise instances satisfying this criterion, which are often chosen to illustrate the computational advantages of new Ising machines. Significant fractions of polynomially simple instances are further found for a wide range of small size models from spin glasses to maximum cut problems. An attractive opportunity to show the advantageous performance of one system over others becomes a demonstration of the platform’s ability to optimise non-deterministic polynomial time (NP-hard) problems that are computationally intractable for the traditional von Neumann architecture machines. Selection of the hardest instances within NP-hard classes could be the key to determining the computational advantages of small and medium-size simulators and may lead to a reliable generalisation of their optimisation performance to a larger scale. Solving the Ising model is NP-hard problem in general, with computational hardness proven for certain coupling matrices[18]

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