Ising machines as hardware solvers of combinatorial optimization problems

Abstract

Ising machines are hardware solvers which aim to find the absolute or approximate ground states of the Ising model. The Ising model is of fundamental computational interest because it is possible to formulate any problem in the complexity class NP as an Ising problem with only polynomial overhead. A scalable Ising machine that outperforms existing standard digital computers could have a huge impact for practical applications for a wide variety of optimization problems. In this review, we survey the current status of various approaches to constructing Ising machines and explain their underlying operational principles. The types of Ising machines considered here include classical thermal annealers based on technologies such as spintronics, optics, memristors, and digital hardware accelerators; dynamical-systems solvers implemented with optics and electronics; and superconducting-circuit quantum annealers. We compare and contrast their performance using standard metrics such as the ground-state success probability and time-to-solution, give their scaling relations with problem size, and discuss their strengths and weaknesses. Key points• Dedicated hardware solvers for the Ising model are of great interest due to the many potential practical applications and the end of Moore's law which motivate alternative computational approaches.• Three main computing methods that Ising machines employ are classical annealing, quantum annealing, and dynamical system evolution. A single machine can operate based on multiple computing approaches.• Today, Ising hardware based on classical digital technologies are the best performing for common benchmark problems. However, the performance is problem dependent and alternative methods can perform well for particular classes of problems.• For particular crafted problem instances, quantum approaches have been observed to have a superior performance over classical algorithms, motivating quantum hardware approaches and quantum-inspired classical algorithms.• Hybrid quantum-classical and digital-analog algorithms are promising for future development; they may harness the complementary advantages of both.

A Finite-Range Search Formulation of Maximum Likelihood MIMO Detection for Coherent Ising Machines

“…We use the real valued equivalent of the MMSE estimate (x m ) as the guess. This step has extremely low computation complexity and can be calculated from the MMSE solution using (4). Given the solution variable u in the realvalued equivalent of the ML-MIMO problem (5), we define…”

“…Several computation technologies like Quantum Annealing (QA), Coherent Ising Machines (CIM), Oscillator-based Ising Machines (OIM), etc. have been proposed in the literature and have shown promising results for solving several challenging computational problems [4]. The Maximum-Likelihood MIMO detection (ML-MIMO) is known to be NP-Hard, and the complexity of optimal methods scales exponentially with the size of the system, making them infeasible for real-life scenarios.…”

Coherent-cluster-state generation in networks of degenerate optical parametric oscillators

“…The coherent Ising machine (CIM) [37][38][39][40][41][42][43][44][45][46][47] stands for optical hardware with the simulation of many-body spin systems as a target application. Similar to platforms for the generation of optical cluster states, the CIM describes a network composed of degenerate optical parametric oscillators (DOPOs).…”

Role of ambient temperature in modulation of behavior of vanadium dioxide volatile memristors and oscillators for neuromorphic applications

“…Consequently, the external temperature will also impact ONN dynamics, affecting ONNs as analog computing architecture. For instance, the process of finding the ground-state of Ising machines can be implemented by gradually lowering the effective temperature (annealing) 24,25 , where such ground-state corresponds to the solution of optimization problems. Recently, an Ising Hamiltonian solver based on VO 2 oscillators has been demonstrated 26 .…”