Optimality and annealing path planning of dynamical analog solvers

Abstract

Recently proposed analog solvers based on dynamical systems, such as Ising machines, are promising platforms for large-scale combinatorial optimization. Yet, given the heuristic nature of the field, there is very limited insight on optimality guarantees of the solvers, as well as how parameter schedules shape dynamics and outcomes. Here, we develop a dynamical mean-field framework to analyze Ising-machine dynamics for finding the ground state energy of the Sherrington-Kirkpatrick(SK) model of spin glasses and identify mechanisms that enable rapid convergence to provenly near-optimal energies. For a fixed target energy density Ec, we show that solutions are typically reached within O(1) matrix vector multiplications, indicating constant time complexity. We further delineate theoretical limitations arising from different parameter-scheduling trajectories and demonstrate a pronounced benefit of temperature-only annealing for the Coherent Ising Machine. Building on these insights, we propose a general framework for designing optimized parameter schedules, thereby improving the practical effectiveness of Ising machines for complex optimization tasks. The superior performance of the dynamical solvers is illustrated by the attainment of the ground state energy of the SK model.

Figures (11)

• Figure 1: (a) Comparison of the diagonal terms of the correlation function $Q$ and the variance of soft spins measured in CIM experiments. (b) Comparison of the decoded energy obtained from Eq. (\ref{['eq:decoded_energy']}) with experimental results from the CIM. The dashed line represents the ground state energy of the SK model in the thermodynamic limit. (c) Dependence of the response function $\partial x(t)/\partial h(t')$ on the value of soft spins $x(t)$. All above simulations were executed using the Euler-Maruyama method with a step size 0.02, $a = 0$, $T=1e-5$, $N=2000$. (d) SimCIM’s dependence of the response function $\partial x(t)/\partial h(t')$ on the value of the soft spins $x(t)$.
• Figure 2: The decoded energy contours and effective gap lines of the CIM. Path 1 represents the conventional strategy of adjusting the parameter $a$, while Path 2 represents annealing solely by temperature. Both paths aim at the same termination point (the red dot at high $a$ and $T$) but are prematurely halted by the effective gap. Notably, path 2 reaches a lower decoded energy than path 1 when they are halted at the same effective gap contour.
• Figure 3: An illustration of the distributional evolution of CIM. Panel (a) fixes T = 0.03 and anneals a; panel (b) fixes a = 0.5 and anneals T.
• Figure 4: Comparison of decoded energy profiles obtained via different parameter-tuning strategies across varying time scales. Data points represent the mean value of the decoded energy from the final step across 20 independent runs with the same annealing protocol for SK models of $N= 2000$. Detailed parameter settings can be found in Appendix \ref{['appendx: Experiments_Setting']}. The dashed line represents the ground truth. $a_e$ marks the terminal point of $a$. Panel (a) corresponds to the gapless regime, (b) aligns with the gap opening at zero temperature, and (c) and (d) represent gap-opened regime. Shaded areas indicate the range of maximum and minimum values observed.
• Figure 5: Comparison of decoded energy profiles obtained via different parameter-tuning strategies across varying time scales for simCIM. Each data point consists of 20 independent trials conducted under different total running time scales, as represented on the $x$-axis.