Soft vector spins with dimensional annealing for combinatorial optimization

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.

Figures (5)

• Figure 1: Example of $d=3$ soft vector spins traversing the Ising energy landscape. For each point in time we evaluate the vector spin energy, $E_\mathrm{vec}$, via $\bm{s}_i = \bm{x}_i / \norm{\bm{x}_i}$, and the Ising energy, $E_\mathrm{Ising}$, via the method in \ref{['sub:proj']}. Solid colored lines show the resulting trajectories for $E_{\mathrm{vec}}$ using different dimensional annealing methods (described in detail later in the text); the dashed lines show the ones for $E_{\mathrm{Ising}}$. For comparison, the evolution of a $d=1$ soft Ising spin system is shown in black (in one dimension $E_{\mathrm{vec}}$ and $E_{\mathrm{Ising}}$ coincide). Vertical dashed lines show the time period where dimensional annealing is active, while the dotted horizontal line shows the Ising ground state energy. Within this region, the dimensional annealing process enables jumps over Ising energy barriers that are not possible in one dimension.
• Figure 2: Trajectories in a $d=2, N=16$ soft vector spin system for different dimensional annealing methods and linear annealing schedules. Note that $t_{\mathrm{f}} = 2 \times 10^3$ in this case. The couplings matrix in this case comes from the WPE. Each line shows the trajectory of a single spin with lighter colours signifying increasing times. Faded out lines in the background show trajectories obtained from other initial conditions and the same coupling matrix. Initial and final states are marked with black and pale yellow dots respectively. Note also that the set of initial conditions is the same for all three methods.
• Figure 3: Success probabilities for finding ground states of "easy", "medium", and "hard" problems using soft vector spins in dimensions $d=1,\dots,5$ and linear annealing schedules. The horizontal axes show the 100 different instances of each problem class, while the vertical axes show the success probability for different dimensional annealing methods. To aid visualization, instances are sorted on the horizontal axis in order of the success rate of $d=2$ AGA.
• Figure 4: Success probabilities for finding ground states of "easy", "medium", and "hard" problems using soft vector spins in dimensions $d=1,\dots,5$ and a feedback mechanism to control gains. The horizontal axes show the 100 different instances of each problem class, while the vertical axes show the success probability for different dimensional annealing methods. To aid visualization, instances are sorted on the horizontal axis in order of the success rate of $d=2$ AGA.
• Figure 5: Scaling of success probabilities with $t_{\mathrm{f}}$ on "easy" problems. We calculate the success probability over 200 different initializations of each method for 11 values of $t_{\mathrm{f}}$ ranging from $1$ to $2 \times 10^3$, comparing AGA, MA, and GCPP in $d=3$ dimensions to just performing gain annealing in $d=1$ dimension. All of this is done for both linear and feedback-driven gain annealing schedules.