OIM: Oscillator-Based Ising Machines for Solving Combinatorial Optimisation Problems

Abstract

We present a new way to make Ising machines, i.e., using networks of coupled self-sustaining nonlinear oscillators. Our scheme is theoretically rooted in a novel result that establishes that the phase dynamics of coupled oscillator systems, under the influence of subharmonic injection locking, are governed by a Lyapunov function that is closely related to the Ising Hamiltonian of the coupling graph. As a result, the dynamics of such oscillator networks evolve naturally to local minima of the Lyapunov function. Two simple additional steps (i.e., adding noise, and turning sub-harmonic locking on and off smoothly) enable the network to find excellent solutions of Ising problems. We demonstrate our method on Ising versions of the MAX-CUT and graph colouring problems, showing that it improves on previously published results on several problems in the G benchmark set. Our scheme, which is amenable to realisation using many kinds of oscillators from different physical domains, is particularly well suited for CMOS IC implementation, offering significant practical advantages over previous techniques for making Ising machines. We present working hardware prototypes using CMOS electronic oscillators.

Analog Coupled Oscillator Based Weighted Ising Machine

“…In this paper, we build upon this initial work and demonstrate a 4 node, fully-connected, differential LC (inductor-capacitor) oscillator based analog circuit with standard electronic components which accurately maps to the Ising model. One key difference between the circuit presented here and that demonstrated previously 17,26 involves the interconnection architecture and oscillator coupling scheme. Whereas previous implementations used a simple resistor to apply the connection weights directly, the circuit demonstrated in this work implements gain ratios in a cross-bar architecture.…”

“…It has been shown previously that these graphs can be represented by a network of coupled nonlinear oscillators whose phase dynamics are described by the Kuramoto model 29 , and that this model maps directly to the Ising Hamiltonian if the phases of these oscillators take values of either 0° or 180°1 7 . One mechanism used to polarize the phases is to introduce an injection-locking signal at twice the natural frequency of the oscillators 17,30 . This 'super-harmonic' injection locking signal, and its application in the context of LCoscillator systems is mathematically similar to the case of the degenerate optical parametric oscillator used in previous optical Ising machines 12,13 , where optical pump pulses at twice the optical oscillation frequency are used to create binary phase values.…”

“…As pointed out previously, this signal is required to extract sensible answers using large node count systems. 17 A high sampling rate phase measurement of the phase transition of the oscillators is shown in Fig. 4.…”

“…Recently, alternative classical methods to solve the Ising model have emerged using optoelectronic parametric oscillators [12][13][14] , memristor cross-bar arrays 11,15,16 , electronic oscillators 17,18 , and GPU based algorithms 19,20 . An analysis of one such coupled oscillator system revealed the potential for a significant speedup over digital computing algorithms at large node sizes 21 .…”

“…In a similar vein, the all-electronic oscillator concept initially proposed by Wang and Roychowdhury introduces the tantalizing prospect of creating a similar system using readily available electronic components interconnected in a parallel fashion and is particularly well suited for chip-scale integration and scaling using present day technologies. 17 In this case, the 'p-bit' is represented by the phase of an oscillator, which can probabilistically settle to one of two values. Previous reports have experimentally demonstrated a Chimera-graph architecture with a total of 240 nodes and described its performance on 20 random problems.…”