Complexity continuum within Ising formulation of NP problems

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.

Order-of-magnitude differences in computational performance of analog Ising machines induced by the choice of nonlinearity

“…This again indicates a link between problem hardness and susceptibility to amplitude inhomogeneity. For uniform node densities and non-random connections on the other hand, which typically contain more easy instances [44], we observe a lower susceptibility to amplitude inhomogeneity.…”

“…For various problems, improvements of up to three orders of magnitude in the time-to-solution are obtained over the polynomial model. The largest differences are observed for graphs with a random connectivity and non-uniform node density, which can generally be considered to contain more difficult instances [44]. This again indicates a link between problem hardness and susceptibility to amplitude inhomogeneity.…”

“…We attribute this to the varying difficulty of the different graphs contained in the Biq Mac library. For randomly generated graphs with spin numbers limited to N = 100, there is a rather high probability of generating easy instances that can be solved in polynomial time [44]. We expect these easy instances to be more robust against faulty mapping due to amplitude inhomogeneity, while the differences in computational performance are more pronounced for difficult problems.…”

Analog Photonics Computing for Information Processing, Inference, and Optimization

“…Calculating the principal eigenvector is also required in other fields such as social network analysis, bibliometrics, recommendation systems, DNA sequencing, bioinformatics, and distributed computing systems. [ 194–196 ]…”

Multidimensional hyperspin machine

“…This means that the PO network deterministically solves the selected optimization problem when driven above the threshold. Such a phenomenology allows to conclude that the optimization problem belongs to the polynomial (P) class of computational complexity [49,65], because finding the ground state of the D-vector Hamiltonian reduces to finding the eigenvector of C with maximal eigenvalue. This is indeed the case of panels a,e and c,g in Fig.…”