OIM: Oscillator-based Ising Machines for Solving Combinatorial Optimisation Problems (1903.07163v1)

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 sub-harmonic 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.

• The paper introduces Oscillator-based Ising Machines (OIM), a classical hardware approach solving combinatorial optimization by mapping problems onto coupled oscillator phase dynamics.
• OIM uses sub-harmonic injection locking and noise injection to minimize a Lyapunov function equivalent to the Ising Hamiltonian via oscillator phase dynamics.
• OIM achieves strong results on MAX-CUT benchmarks and demonstrates potential advantages in scalability and CMOS compatibility over other Ising machines.

Oscillator-based Ising Machines (OIM) present a novel hardware approach to tackle combinatorial optimization problems by employing networks of coupled, self-sustaining nonlinear oscillators (OIM: Oscillator-based Ising Machines for Solving Combinatorial Optimisation Problems, 2019).

OIM Definition: OIM is a hardware realization of Ising machines that exploits the dynamics of coupled oscillators. The fundamental principle relies on the fact that the phase dynamics of these oscillators, particularly when subjected to sub-harmonic injection locking (SHIL), are governed by a Lyapunov function that closely mirrors the Ising Hamiltonian.

Operational Mechanism:

1. Coupled Oscillator Network: OIM utilizes a network of nonlinear oscillators where the coupling strength between individual oscillators is analogous to the weights in the Ising model.
2. Lyapunov Function Minimization: The temporal evolution of the oscillator phases is dictated by a Lyapunov function. As the system evolves, this function decreases, causing the system to settle into local minima.
3. Sub-Harmonic Injection Locking (SHIL): To achieve binarization of the oscillator phases, representing the +1 or -1 spin states, a second harmonic signal (SYNC) is injected, inducing SHIL. This mechanism forces the oscillator phases to stabilize near 0 or $\pi$ radians.
4. Ising Hamiltonian Equivalence: Under SHIL, the Lyapunov function closely approximates the Ising Hamiltonian specifically when the phase values are near 0 or $\pi$. Consequently, the dynamics of the oscillator network effectively minimize a continuous version of the Ising Hamiltonian.
5. Local Minima Escape: To enhance the solution search, noise is introduced, and the SYNC signal amplitude is smoothly varied (ramped up and down). This allows the system to escape from local minima, exploring the energy landscape for lower energy states.

Underlying Principles:

• The Generalized Adler model is used to model the behavior of a single oscillator when subjected to external perturbations.
• A variant of the Kuramoto model captures the phase dynamics of the coupled oscillators.
• The system inherently minimizes a global Lyapunov function, representing an "energy-like" quantity.

Oscillator Types:

• The paper showcases functional hardware prototypes constructed using CMOS electronic oscillators. The architecture can be implemented using various types of oscillators from different physical domains, including CMOS, optical, MEMS, biochemical, and spintronic oscillators.
• Specifically, LC oscillators are used in breadboard implementations.

Applications and Performance Metrics:

• OIM is applied to solve NP-complete combinatorial optimization problems such as the MAX-CUT and graph coloring problems.
• In MAX-CUT benchmark tests (G-set), OIM identifies the best-known cut values for numerous problems and even surpasses previously published results in certain instances.
• Preliminary studies suggest that the computation time of OIM scales slowly with the number of spins, indicating a potential advantage over simulated annealing as the hardware is scaled up.

Comparative Analysis:

• OIM is benchmarked against Coherent Ising Machines (CIM), D-Wave quantum Ising machines, and CMOS hardware-accelerated simulated annealing chips.
• In contrast to CIM and D-Wave, OIM is purely classical scheme, making it suitable for CMOS implementation, offering benefits in terms of scalability, cost-effectiveness, and integration capabilities.
• Compared to hardware-based simulated annealing, OIM exhibits resilience to variability due to the ability to calibrate oscillator frequencies. Additionally, the continuous/analog nature of OIM provides the potential for faster computation times.