Computational complexity continuum within Ising formulation of NP problems

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. Giv...

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Veröffentlicht in:Communications physics Jg. 5; H. 1; S. 1 - 10
Hauptverfasser: Kalinin, Kirill P., Berloff, Natalia G.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: London Nature Publishing Group UK 12.01.2022
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Algorithms
Complexity
Criteria
Electronic systems
Graphs
Ising model
Linear programming
Numerical methods
Optimization
Performance evaluation
Physics
Physics and Astronomy
Rewiring
Simulators
Spin glasses
ISSN:2399-3650, 2399-3650
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Zusammenfassung: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. The advantage of unconventional computing architectures is commonly demonstrated by solving an NP-hard problem, but some instances are easier to solve than others. Here, an optimisation simplicity criterion is proposed that classifies the complexity of instances on optical or electronic neuromorphic computers.
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ISSN:2399-3650
2399-3650
DOI:10.1038/s42005-021-00792-0