More efficient directed-loop solutions

Determine whether a directed-loop solution for the three-operator stochastic series expansion algorithm can achieve greater simulation efficiency than the solution obtained by minimizing the probability of no-update events.

Background

The paper formulates directed-loop equations for an SSE algorithm that treats the diagonal magnetic-field operator as an independent operator class alongside the diagonal and off-diagonal Heisenberg operators. These equations admit infinitely many solutions satisfying detailed balance and normalization.

For the simulations, the authors select a particular solution that minimizes the probability of no-update events, motivated by the expectation that this may improve efficiency. They explicitly acknowledge that this criterion need not produce the most efficient solution, leaving open whether another admissible choice of directed-loop probabilities can reduce autocorrelation times further.

References

This choice is based on the intuitive expectation that suppressing such event may improve the simulation efficiency. However, it cannot be excluded that a more efficient directed-loop solution exists in which this probability is not minimized.

— Decoupling the Magnetic Field Operator as an Independent Operator Class in Stochastic Series Expansion Quantum Monte Carlo  (2609.23978 - Jin et al., 21 Sep 2026) in Section 4, “Directed-loop Equation”