Bryan K. Clark
University of Illinois Urbana-Champaign

Stefano Gandolfi
Los Alamos National Laboratory

Francesco Pederiva
University of Trento - INFN

Martin J. Savage
INT, University of Washinghton

Diversity Coordinator:

Martin J. Savage
INT, University of Washinghton

Program Coordinator:

Farha Habib
(206) 685-4286

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For full consideration, please apply by May 18, 2018.

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INT Program INT-18-2b

Advances in Monte Carlo Techniques for Many-Body Quantum Systems

July 30 - September 7, 2018


Quantum Monte Carlo (QMC) techniques have become essential tools in a myriad of fields including computational condensed matter, high-energy physics, lattice qcd, quantum chemistry and nuclear theory. Although the physics that is being extracted is diverse in nature, the methodologies, algorithms and fundamental obstacles are common across these fields. In each case, there exists an effective Hamiltonian (Lagrangian) whose ground state (partition function) properties are being computed.

Unfortunately, the computational complexity for current exact algorithmic approaches scale exponentially with system size. In the context of stochastic algorithms (i.e. projector quantum Monte Carlo) this problem comes from the decay of the signal to noise ratio occurring in the computation of highly dimensional integrals in which the integrand has alternate signs. This is typically called the fermion sign problem. This exponential scaling, though, arises in various ways in different algorithms. In exact diagonalization, the wave function requires exponential memory to store; in DMRG the bond dimension grows with the entanglement (exponentially with the width of the system); and in diagrammatic approaches, the system size is infinite but there is an exponential scaling in order.

Historically, this problem has been dealt with by limiting calculations to small system sizes (or zero baryon density) or by using uncontrolled approximations such as using the variational principle to 'guess' a good ansatz for the solution, or by applying an artificial constraint on the space that can be explored by a random walk. Within the last five years, there has been an algorithmic paradigm shift in approaching strongly correlated systems. Various new systematically exact algorithms are being developed which still scale exponentially but do so with reduced computational complexity (i.e a smaller exponent or exponential cost in a different parameter). Examples include full configuration interaction quantum Monte Carlo (FCIQMC) which is exponential in system size but not temperature, diagrammatic Monte Carlo which is exponential in diagram order but not system size, two-dimensional tensor networks such as the projected entangled plaquette state (PEPS) which is a variational wave function and which compactly represents many ground states but is exponential in cost to evaluate, etc. In addition, various hybrid algorithmic approaches have been developed to take simultaneous advantage of features of multiple algorithms such as semi-stochastic QMC, partial node FCIQMC and QMC+MPS approaches. It should be noted that this change in perspective has been driven by various forces including significantly improved computational resources which makes low scaling exponential algorithms potentially useful.

This workshop then exists within this context of significant new algorithmic development and possibilities in the area of quantum Monte Carlo techniques broadly defined. In particular, we believe that there is significant room for new algorithms, unique hybrid approaches, and order of magnitude improvements in newly developed techniques. As an example of the confluence of these ideas, we consider the specific case of nuclear physics. The recent progress in the field concerns a better understanding of this general formulation of MC methods in quantum systems. For example, in the field of nuclear structure calculations, it became possible in the last few years to study properties of nuclei and nuclear matter using nuclear interactions derived within chiral effective field theory. This was possible by adapting EFT Hamiltonians to be used by QMC methods on the continuum, but also through the development of novel algorithms that work in momentum and/or Fock space, like configuration interaction Monte Carlo (CIMC), and permit solutions of the nuclear many-body problem using non-local Hamiltonians.

These advances are extremely important, as it is possible now to study and quote theoretical uncertainties given by using different models. In addition, recent calculations showed that QMC methods can also be used to calculate response functions and matrix elements to calculate, for example, electron- and neutrino-scattering, electroweak transitions; they will also be used to calculate ab initio important quantities like matrix elements related to double-beta decays and neutrino rates in neutron stars and supernovae. Historically, cross-fertilization of ideas among communities dealing with different physical problems have proven particularly fruitful. Consider as examples the transition of the CORE approach from its origin in lattice QCD to the condensed matter community; the utilization of coupled cluster theory in nuclear theory from its origin in quantum chemistry and more recently its variational application to condensed matter systems; the application of annihilation techniques originally devised in the condensed matter community to applications in quantum chemistry as an important component of FCIQMC.

We note five key themes in the purview of this workshop:

The proposed program bears a strong connection to several recent INT programs, in particular the previous program on Advances in Quantum Monte Carlo Techniques for Non-Relativistic Many-Body Systems (INT-13-2a). In this sense, we expect this program to play an important role in sustaining the communication of methods across fields, extending it more explicitly to the LQCD community, as well as maintaining the continued discussion and interest in the development of stochastic methods for the many-body problem within nuclear physics and beyond.

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