Soft vector spins with dimensional annealing for combinatorial optimization

TL;DR

This paper introduces a model of soft vector spins with dimensional annealing to enhance the performance of analog hardware solving combinatorial optimization problems, demonstrating improved results with higher-dimensional degrees of freedom.

Contribution

The work presents a new vector spin model and three dimensional annealing methods, showing how vectorial degrees of freedom improve Ising energy minimization on benchmark problems.

Findings

Vectorial degrees of freedom outperform one-dimensional spins in finding ground states.

Advantages are most significant for dimensions greater than or equal to 3.

Diminishing returns observed as the number of dimensions increases further.

Abstract

Recently, purpose-built analog hardware that can efficiently minimize the Ising energy and thereby solve a variety of combinatorial optimization problems has been receiving widespread attention. In this work, we show how multidimensional, vectorial degrees of freedom, that are either naturally present or can be artificially created in such hardware, could strengthen the capability to find optimal solutions to optimization problems. In order to achieve this, we introduce a simple model of soft vector spins that should be implementable on a variety of analog hardware platforms as well as three different dimensional annealing methods which harness the enlarged phase space of the vectorial degrees of freedom to minimize the Ising energy. We perform simulations on different benchmark problems and show that for all dimensional annealing methods we tested, vectorial degrees of freedom improve upon one-dimensional degrees of freedom when it comes to finding the ground state of the Ising model. In particular, we find that this advantage becomes most pronounced for $d \gtrsim 3$ dimensional degrees of freedom, with diminishing returns as the dimension is increased further. Our results could inspire new analog optimization hardware and algorithms that explicitly incorporate the advantage of vectorial degrees of freedom.