Reducing hyperparameter sensitivity in measurement-feedback based Ising machines

TL;DR

This paper analyzes the discrepancy in the range of effective hyperparameters in measurement-feedback architectures of analog Ising machines, and proposes and experimentally verify a method to reduce the sensitivity to hyperparameter selection of these measurement-feedback architectures.

Abstract

Analog Ising machines have been proposed as heuristic hardware solvers for combinatorial optimization problems, with the potential to outperform conventional approaches, provided that their hyperparameters are carefully tuned. Their temporal evolution is often described using time-continuous dynamics. However, most experimental implementations rely on measurement-feedback architectures that operate in a time-discrete manner. We observe that in such setups, the range of effective hyperparameters is substantially smaller than in the envisioned time-continuous analog Ising machine. In this paper, we analyze this discrepancy and discuss its impact on the practical operation of Ising machines. Next, we propose and experimentally verify a method to reduce the sensitivity to hyperparameter selection of these measurement-feedback architectures.

Figures (11)

• Figure 1: Basic working principle of an analog IM.(a) Schematic illustration of the operation of an analog IM. Both analog noise, which helps the system escape local minima, and a feedback signal, which drives the system toward lower energy levels, are essential for all analog IM implementations. The feedback signal can be realized in several ways, some of which are shown in (b). (c) shows an example simulation result of a time-discrete analog IM solving the $g05\_80.1$ MaxCut problem. The upper panel displays the simulated spin amplitudes, while the lower panel shows the corresponding energy evolution.
• Figure 2: Comparison of the brute force parameter grid scan of a time-discrete and a time-continuous analog IM, for the g05$\_$80.1-benchmark problem. Both figures have coupling strength $\beta$ and the gain $\alpha$ on the x- and y-axis respectively, and the transient success rate (TSR) color coded. For the time-discrete system shown in (a), the parameter range leading to non-zero TSR is significantly smaller then for the time-continuous system shown in (b). The inset in (a) shows a high-resolution zoom-in of the area indicated by the red dashed box.
• Figure 3: Influence of the h-value on the parameter space.a The brute force parameter grid scan of the g05$\_$80.1 problem for four different h-values. The x- and y-axes of the grid scans are the coupling strength $\beta$ and the gain $\alpha$, respectively. The color bar indicates the transient success rate (TSR), while a zero-valued TSR is represented by the grey color. b The area of operation (AOO), defined as the percentage of ($\alpha$,$\beta$)-combinations that results in a non-zero transient success rate (TSR), as a function of the h-value for different problem sizes. Each problem size has a separate symbol and color. The color shaded areas represent the different problem instances of a specific problem size.
• Figure 4: Schematic of the time-multiplexed opto-electronic CIM. In this hybrid setup, the input signal is optically modulated, then converted into an analog voltage and finally sampled, digitized, and stored on an FPGA. The calculated feedback signal is converted back into a voltage and sent to the modulation input of the MZM.
• Figure 5: Experimental verification of the decreasing parameter sensitivity with decreasing $h$-value.(a) Parameter scans over the normalized $\alpha_{\mathrm{norm}}, \beta_{\mathrm{norm}}$-parameter space for the g05_80.1 benchmark problem, shown for five different values of $h$. The evaluated metric is the transient success rate (TSR). As $h$ decreases, a larger region of the hyperparameter space yields a non-zero TSR. (b) Resulting AOO values corresponding to the six evaluated $h$-values are indicated by symbols (line is intended to guide the eye).