Majorana sign problem in determinant quantum Monte Carlo

In determinant quantum Monte Carlo simulations, the trace of the evolution operator, constructed using Fermionic Gaussian operators, serves as a vital sampling weight. The sign of this trace, responsible for the well-known sign problem, has been a subject of extensive research for decades. Tremendous efforts have been made to address sign problems by leveraging symmetries or special mathematical structures. However, in majorana systems lacking these special requirements, determining the sign of the evolution operator constructed by Majorana Gaussian operators has posed a longstanding challenge. Previously, the trace could only be determined with an ambiguous sign. In this study, we successfully resolved this ambiguity and derived a closed-form formula for the trace using Pfaffian techniques, enabling complete trace calculation in polynomial time for the first time. Additionally, we explored the overlap of Hartree-Fock-Bogoliubov states evolved by Majorana Gaussian operators. Our findings represent a significant advancement, shedding light on the development in Majorana quantum Monte Carlo simulations and offering new possibilities for the discovery of new sign problem-free models exhibitting novel quantum phases of matter.