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

TL;DR

This paper introduces a fast-update technique for self-learning Monte Carlo algorithms in quantum impurity models, significantly reducing computational complexity and achieving exponential speedup, especially at low temperatures.

Contribution

The authors develop a novel polynomial-based update method that removes CPU time dependence on the number of vertices, enabling exponential speedup in quantum Monte Carlo simulations.

Findings

CPU time per update reduced from O(nm) to O(m^2)

Achieves exponential speedup over existing methods

Effective at low temperatures with logarithmic n-dependence

Abstract

We propose a novel technique for speeding up the self-learning Monte Carlo method applied to the single-site impurity model. For the case where the effective Hamiltonian is expressed by polynomial functions of differences of imaginary-time coordinate between vertices, we can remove the dependence of CPU time on the number of vertices, $n$, by saving and updating some coefficients for each insertion and deletion process. As a result, the total cost for a single-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 the existing algorithms, in which the absolute value is used instead of the difference as the variable of polynomial functions, we can limit the CPU time for a single step of Monte Carlo update to $O(m^2 + m \log n)$ with the help of balanced binary search trees. We demonstrate that our proposed algorithm with only logarithmic $n$-dependence achieves an exponential speedup from the existing methods, which suffer from severe performance issues at low temperatures.