Abstract : Chemistry, physics, and materials science have benefited tremendously from advances in algorithmic tools for the simulation of quantum systems. In recent years, ideas developed in collaboration with the applied mathematics community have played an increasingly prominent role. This minisymposium will focus on recent algorithmic advances in computational quantum mechanics driven by numerical linear algebra, numerical methods for partial differential equations and integral equations, fast algorithms for the manipulation of structured operators, convex optimization, tensor networks, randomized algorithms, and machine learning methods.
[04860] Deterministic algorithms for the efficient evaluation of Feynman diagrams
Format : Talk at Waseda University
Author(s) :
Jason Kaye (Flatiron Institute, Simons Foundation)
Denis Golež (Jožef Stefan Institute)
Hugo U. R. Strand (Örebro University)
Abstract : The evaluation of Feynman diagrams is a fundamental computational task, and bottleneck, in Green's function methods for quantum many-body calculations. Mathematically, diagrammatic expressions take the form of high-dimensional integrals involving products of low-dimensional functions. Numerical methods for the efficient evaluation of these diagrams are predominantly based on Monte Carlo or quasi-Monte Carlo sampling. I will discuss recent progress in algorithms which exploit the structure of the diagrammatic expressions to obtain efficient deterministic algorithms. In particular, I will describe an algorithm for the evaluation of imaginary time diagrams based on sum-of-exponentials representations of Green's functions, and mention a related algorithm based on the decomposition of integrands into tensor trains using cross interpolation.
Abstract : In this talk, we apply quantics tensor train (QTT) to numerical simulations in quantum field theories$^1$. Using QTT, we compress space-time dependence of correlation functions and solve equations in the compressed form. We also introduce quantics tensor cross interpolation to enhance the power of QTT-based calculations$^2$. 1. H. Shinaoka et al., arXiv:2210.12984v2 (to appear in Phys. Rev. X), 2. M. K. Ritter et al., including H. Shinaoka, arXiv:2303.11819v2.
[03608] Circuit Forms of Compressed Functions and Operators with Applications
Format : Talk at Waseda University
Author(s) :
Edwin Miles Stoudenmire (Flatiron Institute)
Abstract : Recently there has been growing interest in representing functions as tensor networks using the "quantized tensor train" format, which enables algorithms (integration, function optimization, ...) typically scaling logarithmically in the grid size. I will discuss techniques to construct functions and operators in this format using non-unitary analogues of quantum circuits. This leads to fast algorithms for Fourier and wavelet transforms, and has promising applications for systems of continuous variables in classical and quantum physics.
[05166] Tensor-network sketching and many-body physics
Format : Talk at Waseda University
Author(s) :
Yuehaw Khoo (The University of Chicago)
Yoonhaeng Hur (The University of Chicago)
Yian Chen (The University of Chicago)
Jeremy Hoskins (The University of Chicago)
Michael Lindsey (UC Berkeley)
Edwin Miles Stoudenmire (Flatiron Institute)
Abstract : We study how a function can be estimated as a low-rank tensor-network from Monte-Carlo samples, without the use of optimization and without the curse of dimensionality. We demonstrate the usefulness in generative modeling and many-body physics.
[03946] Analyzing Non-equilibrium Quantum Many-body Dynamics by Dynamic Mode Decomposition
Format : Talk at Waseda University
Author(s) :
Chao Yang (Lawrence Berkeley National Lab)
Jia Yin (Lawrence Berkeley National Lab)
Yuanran Zhu (Lawrence Berkeley National Lab)
Abstract : A practical way to compute time-dependent observables of an out-of-equilibrium quantum many-body system is to focus on the single-particle Green's function defined on the Keldysh contour. The equation of motion satisfied by such a Green's function is a set of nonlinear integro-differential equations called the Kadanoff-Baym equations. We will describe numerical methods for solving these equations and show how to use dynamic mode decomposition to reduce their computational complexity.
[05317] Some mathematical insights on DMET
Format : Talk at Waseda University
Author(s) :
Fabian Maximilian Faulstich (Rensselaer Polytechnic Institute)
Abstract : High-accuracy methods are crucial for simulating static correlated systems, but they often scale severely. DMET can solve this by scaling highly accurate solvers. This talk shows that the exact ground-state density matrix is a fixed point of DMET for non-interacting systems, with a unique physical solution in the weakly-interacting regime. DMET is exact to first order in the coupling parameter, and numerical simulations confirm these results. Assumptions behind their validity are also discussed.
[04977] Efficient algorithms for Brillouin zone and frequency integration
Format : Talk at Waseda University
Author(s) :
Lorenzo Xavier Van Munoz (Massachusetts Institute of Technology)
Jason Kaye (Flatiron Institute, Simons Foundation)
Sophie Beck (Flatiron Institute, Simons Foundation)
Abstract : Brillouin zone integration is a standard operation in electronic structure calculations used to compute numerous physical observables. For integrands with broad features at scale $\eta$, standard equispaced integration methods are highly effective. However, when $\eta$ is small, adaptive methods become necessary to achieve converged results. We extend these adaptive, high-order accurate methods to problems with an additional frequency integral, such as the optical conductivity, and discuss how to control the error in these iterated integrals.
[04273] Extraction of resonant states in systems with defects
Format : Talk at Waseda University
Author(s) :
Eloise Letournel (CERMICS)
Antoine Levitt (Université Paris Saclay (LMO))
Luigi Genovese (CEA Grenoble)
Ivan Duchemin (CEA Grenoble)
Simon Ruget (CERMICS)
Abstract : We introduce a numerical method to compute resonances induced by localized defects in crystals. We express the resonance in terms of a “resonance source" strictly localized within the defect region. We then compute a kernel equation, applying against this source the Green's function of the perfect crystal, which we show can be computed efficiently by a complex deformation of the Brillouin zone (BCD).
[03564] Density functional theories: reformulations and regularizations
Format : Talk at Waseda University
Author(s) :
Michael Ruggenthaler (Max-Planck Institute for the Structure and Dynamics of Matter)
Abstract : Density functional theories try to reformulate quantum theories in terms of a set of reduced quantities, turning high-dimensional linear problems into low-dimensional but non-linear problems (1,2,3). In this talk I will give a short overview of open mathematical problems in density functional theories and provide potential solution strategies, either by Moreau-Yosida regularization (4) or by reformulating the the basic mappings (2,3).
(1) Ruggenthaler, M., Penz, M., & Van Leeuwen, R. (2015). Existence, uniqueness, and construction of the density-potential mapping in time-dependent density-functional theory. Journal of Physics: Condensed Matter, 27(20), 203202.
(2) Penz, M., Tellgren, E. I., Csirik, M. A., Ruggenthaler, M., & Laestadius, A. (2022). The structure of the density-potential mapping. Part I: Standard density-functional theory. arXiv preprint arXiv:2211.16627.
(3) Penz, M., Tellgren, E.I., Csirik, M.A., Ruggenthaler, M, & Laestadius, A (2023). The structure of the density-potential mapping. Part II: Including magnetic fields. arXiv preprint arXiv:2303.01357
(4) Penz, M., Laestadius, A., Tellgren, E. I., & Ruggenthaler, M. (2019). Guaranteed convergence of a regularized Kohn-Sham iteration in finite dimensions. Physical Review Letters, 123(3), 037401.
[04839] Fine-grained parallelism is flow-based Monte Carlo algorithms
Format : Talk at Waseda University
Author(s) :
Michael Samuel Albergo (New York University)
Michael S Albergo (New York University)
Abstract : Transport-based generative models have become an active topic of inquiry in scientific computing and Monte Carlo-based numerical methods, with demonstrated applications ranging from lattice quantum field theory to molecular systems. In this will talk, I will discuss the flow-based Monte Carlo approach and present some results related to taking advantage of its embarrassingly parallel setup.
[04906] An unambiguous and robust formulation for Wannier localization
Format : Talk at Waseda University
Author(s) :
Kangbo Li (Cornell University)
Abstract : We provide a new variational definition for the spread of an orbital under periodic boundary conditions (PBCs) that is continuous with respect to the gauge, well-suited to diffuse orbitals, and can be adapted for schemes to compute localized Wannier functions. Existing definitions do not satisfy all these desiderata, partly because they depend on an "orbital center''---an ill-defined concept under PBCs. Moreover, we illustrate a more robust and efficient ($10\times$$-$$70\times$ fewer iterations) localization scheme.
