Ising machines are dedicated hardware solvers of NP-hard optimization problems. However, they do not always find the most optimal solution. The probability of finding this optimal solution depends on the problem at hand. Using continuation methods, we show that this is closely linked to the bifurcation sequence of the optimal solution. From this bifurcation analysis, we can determine the effectiveness of solution schemes. Moreover, we find that the proper choice of implementation of the Ising machine can drastically change this bifurcation sequence and therefore vastly increase the probability of finding the optimal solution.

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