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We propose a novel technique for speeding up the self-learning Monte Carlomethod applied to the single-site impurity model. For the case where theeffective Hamiltonian is expressed by polynomial functions of differences ofimaginary-time coordinate between vertices, we can remove the dependence of CPUtime on the number of vertices, $n$, by saving and updating some coefficientsfor each insertion and deletion process. As a result, the total cost for asingle-step update is drastically reduced from $O(nm)$ to $O(m^2)$ with $m$being the order of polynomials in the effective Hamiltonian. Even for theexisting algorithms, in which the absolute value is used instead of thedifference as the variable of polynomial functions, we can limit the CPU timefor a single step of Monte Carlo update to $O(m^2 + m \log n)$ with the help ofbalanced binary search trees. We demonstrate that our proposed algorithm withonly logarithmic $n$-dependence achieves an exponential speedup from theexisting methods, which suffer from severe performance issues at lowtemperatures.

Fast-update in self-learning algorithm for continuous-time quantum Monte Carlo