Fast Principal Minor Algorithms for Quantum Many Body Systems

Abstract

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.

References in corpus (19)

• Continuous-time Monte Carlo methods for quantum impurity models
• Fermi-Polaron: Diagrammatic Monte Carlo for Divergent Sign-Alternating Series
• Monte Carlo study of two-dimensional Bose-Hubbard model
• Critical Temperature and Thermodynamics of Attractive Fermions at Unitarity
• Tracking the Footprints of Spin Fluctuations: A MultiMethod, MultiMessenger Study of the Two-Dimensional Hubbard Model
• Mott insulating states with competing orders in the triangular lattice Hubbard model
• Sub-matrix updates for the Continuous-Time Auxiliary Field algorithm
• Quantum Monte-Carlo for correlated out-of-equilibrium nanoelectronics devices
• Emergent BCS regime of the two-dimensional fermionic Hubbard model: ground-state phase diagram
• Quantum Quasi-Monte Carlo Technique for Many-Body Perturbative Expansions
• Renormalized perturbation theory at large expansion orders
• High-precision numerical solution of the Fermi polaron problem and large-order behavior of its diagrammatic series
• Quantum Quasi-Monte Carlo algorithm for out-of-equilibrium Green functions at long times
• Real-space cluster dynamical mean-field theory: Center focused extrapolation on the one- and two particle level
• Homotopic Action: A Pathway to Convergent Diagrammatic Theories
• Entropy in the non-Fermi-liquid regime of the doped $2d$ Hubbard model
• Inclusion-Exclusion Principle for Open Quantum Systems with Bosonic Bath
• Efficient One-Loop-Renormalized Vertex Expansions with Connected Determinant Diagrammatic Monte Carlo
• Diagrammatic Monte Carlo Method for Impurity Models with General Interactions and Hybridizations

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