Scaling advantage of chaotic amplitude control for high-performance combinatorial optimization

Timothée Leleu,Ryan Hamerly,Farad Khoyratee,Takashi Kohno,Timothée Levi,Kazuyuki Aihara

doi:10.1038/s42005-021-00768-0

Abstract

The development of physical simulators, called Ising machines, that sample from low energy states of the Ising Hamiltonian has the potential to transform our ability to understand and control complex systems. However, most of the physical implementations of such machines have been based on a similar concept that is closely related to relaxational dynamics such as in simulated, mean-field, chaotic, and quantum annealing. Here we show that dynamics that includes a nonrelaxational component and is associated with a finite positive Gibbs entropy production rate can accelerate the sampling of low energy states compared to that of conventional methods. By implementing such dynamics on field programmable gate array, we show that the addition of nonrelaxational dynamics that we propose, called chaotic amplitude control, exhibits exponents of the scaling with problem size of the time to find optimal solutions and its variance that are smaller than those of relaxational schemes recently implemented on Ising machines.

Highlights

The development of physical simulators, called Ising machines, that sample from low energy states of the Ising Hamiltonian has the potential to transform our ability to understand and control complex systems
Have been used in various computational models that are applied to NP-hard combinatorial optimization problems such as Hopfield-Tank neural networks[45], coherent Ising machines[46], and correspond to the “soft” spin description of frustrated spin systems[47]
In order to test if the nonrelaxational dynamics of chaotic amplitude control might be able to accelerate the search of mean-field dynamics for finding the ground state of typical frustrated systems, we look for the ground states of Sherrington-Kirkpatrick (SK) spin glass instances using the numerical simulation of eqs. (1) to (3) and compare time to solutions with those of two closely related relaxational schemes: noisy mean-field annealing (NMFA)[22] and the simulation of the coherent Ising machine

Summary

Introduction

The development of physical simulators, called Ising machines, that sample from low energy states of the Ising Hamiltonian has the potential to transform our ability to understand and control complex systems. The difficulty in constructing connections between constituting elements of the hardware is often the main limiting factor to scalability and performance for these systems[15,19] These devices often implement schemes that are directly related to the concept of annealing (either simulated[20,21], mean-field[22,23], chaotic[18,24], and quantum17,25) in which the escape from the numerous local minima[26] and saddle points[27] of the free energy function can only be achieved under very slow modulation of the control parameter (see Fig. 1(c)). In order to extend numerical analysis to large problem sizes and limit finite-size effects, we implement a scheme that we name chaotic amplitude control (CAC) on a field programmable gate array (FPGA, see Fig. 1(b)) and show that the developed hardware can be faster for finding these optimal configurations in the limit of large problem sizes than many state-of-the-art algorithms and Ising machines for some reference benchmarks with enhanced energy efficiency

Methods

Results

Conclusion