Fast Principal Minor Algorithms for Quantum Many Body Systems

The computation of determinants plays a central role in diagrammatic Monte Carlo algorithms for strongly correlated systems. The evaluation of large numbers of determinants can often be the limiting computational factor determining the number of attainable diagrammatic expansion orders. In this work we build upon the algorithm presented in [\emph{Linear Algebra and its applications} 419.1 (2006), 107-124] which computes all principal minors of a matrix in $O(2^n)$ operations. We present multiple generalizations of the algorithm to the efficient evaluation of certain subsets of all principal minors with immediate applications to Connected Determinant Diagrammatic Monte Carlo within the normal and symmetry-broken phases as well as Continuous-time Quantum Monte Carlo. Additionally, we improve the asymptotic scaling of diagrammatic Monte Carlo formulated in real-time to $O(2^n)$ and report speedups of up to a factor $25$ at computationally realistic expansion orders. We further show that all permanent-principal-minors, corresponding to sums of bosonic Feynman diagrams, can be computed in $O(3^n)$, making diagrammatic Monte Carlo for bosonic and mixed systems a viable path worth exploring.

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