The Effect of Penalty Factors of Constrained Hamiltonians on the Eigenspectrum in Quantum Annealing

Constrained optimization problems are usually translated to (naturally unconstrained) Ising formulations by introducing soft penalty terms for the previously hard constraints. In this work, we empirically demonstrate that assigning the appropriate weight to these penalty terms leads to an enlargement of the minimum spectral gap in the corresponding eigenspectrum, which also leads to a better solution quality on actual quantum annealing hardware. We apply machine learning methods to analyze the correlations of the penalty factors and the minimum spectral gap for six selected constrained optimization problems and show that regression using a neural network allows to predict the best penalty factors in our settings for various problem instances. Additionally, we observe that problem instances with a single global optimum are easier to optimize in contrast to ones with multiple global optima.