Beyond Walkers in Stochastic Quantum Chemistry: Reducing Error using Fast Randomized Iteration.

We introduce a family of methods for the full configuration interaction problem in quantum chemistry, based on the fast randomized iteration (FRI) framework [L.-H. Lim and J. Weare, SIAM Rev. 59, 547 (2017)]. These methods stochastically impose sparsity during iterations of the power method and can be viewed as a generalization of full configuration interaction quantum Monte Carlo (FCIQMC) without walkers. In addition to the multinomial scheme commonly used to sample excitations in FCIQMC, we present a systematic scheme where excitations are not sampled independently. Performing ground-state calculations on five small molecules at fixed cost, we find that the systematic FRI scheme is about 10 to 60 times more statistically efficient than the multinomial FRI scheme, which is in turn about 1.4 to 200 times more statistically efficient than FCIQMC.

[1]  Jonathan Weare,et al.  Practical rare event sampling for extreme mesoscale weather. , 2019, Chaos.

[2]  Yingzhou Li,et al.  Coordinate Descent Full Configuration Interaction. , 2019, Journal of chemical theory and computation.

[3]  Alex J W Thom,et al.  Exciting Determinants in Quantum Monte Carlo: Loading the Dice with Fast, Low-Memory Weights. , 2018, Journal of chemical theory and computation.

[4]  N. S. Blunt,et al.  Communication: An efficient and accurate perturbative correction to initiator full configuration interaction quantum Monte Carlo. , 2018, The Journal of chemical physics.

[5]  Zhe Wang,et al.  The full configuration interaction quantum Monte Carlo method in the lens of inexact power iteration , 2017, 1711.09153.

[6]  Mario Motta,et al.  Ab initio computations of molecular systems by the auxiliary‐field quantum Monte Carlo method , 2017, 1711.02242.

[7]  Charles J. C. Scott,et al.  Stochastic coupled cluster theory: Efficient sampling of the coupled cluster expansion. , 2017, The Journal of chemical physics.

[8]  Sandeep Sharma,et al.  PySCF: the Python‐based simulations of chemistry framework , 2017, 1701.08223.

[9]  Ali Alavi,et al.  Semistochastic Heat-Bath Configuration Interaction Method: Selected Configuration Interaction with Semistochastic Perturbation Theory. , 2016, Journal of chemical theory and computation.

[10]  Francesco A Evangelista,et al.  A Deterministic Projector Configuration Interaction Approach for the Ground State of Quantum Many-Body Systems. , 2016, Journal of chemical theory and computation.

[11]  John D Chodera,et al.  A Simple Method for Automated Equilibration Detection in Molecular Simulations. , 2016, Journal of chemical theory and computation.

[12]  Martin Head-Gordon,et al.  A deterministic alternative to the full configuration interaction quantum Monte Carlo method. , 2016, The Journal of chemical physics.

[13]  W. A. Vigor,et al.  Understanding and improving the efficiency of full configuration interaction quantum Monte Carlo. , 2016, The Journal of chemical physics.

[14]  C J Umrigar,et al.  Efficient Heat-Bath Sampling in Fock Space. , 2015, Journal of chemical theory and computation.

[15]  C J Umrigar,et al.  Observations on variational and projector Monte Carlo methods. , 2015, The Journal of chemical physics.

[16]  Jonathan Weare,et al.  Fast Randomized Iteration: Diffusion Monte Carlo through the Lens of Numerical Linear Algebra , 2015, SIAM Rev..

[17]  Ali Alavi,et al.  An excited-state approach within full configuration interaction quantum Monte Carlo. , 2015, The Journal of chemical physics.

[18]  George H. Booth,et al.  Semi-stochastic full configuration interaction quantum Monte Carlo: Developments and application. , 2015, The Journal of chemical physics.

[19]  Ali Alavi,et al.  Unbiased reduced density matrices and electronic properties from full configuration interaction quantum Monte Carlo. , 2014, The Journal of chemical physics.

[20]  H. Yao,et al.  Solving the fermion sign problem in quantum Monte Carlo simulations by Majorana representation , 2014, 1408.2269.

[21]  Gustavo E. Scuseria,et al.  Sign problem in full configuration interaction quantum Monte Carlo: Linear and sublinear representation regimes for the exact wave function , 2014, 1407.4800.

[22]  W. A. Vigor,et al.  Minimising biases in full configuration interaction quantum Monte Carlo. , 2014, The Journal of chemical physics.

[23]  Ali Alavi,et al.  Linear-scaling and parallelisable algorithms for stochastic quantum chemistry , 2013, 1305.6981.

[24]  C J Umrigar,et al.  Semistochastic projector Monte Carlo method. , 2012, Physical review letters.

[25]  Martin Hairer,et al.  Improved Diffusion Monte Carlo , 2012, 1207.2866.

[26]  Daniel Foreman-Mackey,et al.  emcee: The MCMC Hammer , 2012, 1202.3665.

[27]  W. Lester,et al.  Quantum Monte Carlo and related approaches. , 2012, Chemical reviews.

[28]  N. S. Blunt,et al.  The sign problem and population dynamics in the full configuration interaction quantum Monte Carlo method. , 2011, The Journal of chemical physics.

[29]  Ali Alavi,et al.  Breaking the carbon dimer: the challenges of multiple bond dissociation with full configuration interaction quantum Monte Carlo methods. , 2011, The Journal of chemical physics.

[30]  Ali Alavi,et al.  A study of electron affinities using the initiator approach to full configuration interaction quantum Monte Carlo. , 2011, The Journal of chemical physics.

[31]  Alex J W Thom,et al.  Stochastic coupled cluster theory. , 2010, Physical review letters.

[32]  Ali Alavi,et al.  Communications: Survival of the fittest: accelerating convergence in full configuration-interaction quantum Monte Carlo. , 2010, The Journal of chemical physics.

[33]  R. Needs,et al.  Continuum variational and diffusion quantum Monte Carlo calculations , 2010, Journal of physics. Condensed matter : an Institute of Physics journal.

[34]  Ali Alavi,et al.  Fermion Monte Carlo without fixed nodes: a game of life, death, and annihilation in Slater determinant space. , 2009, The Journal of chemical physics.

[35]  Peter A. Maksym,et al.  Auxiliary field quantum Monte-Carlo simulation of interacting electrons in quantum dots , 2005 .

[36]  M. Troyer,et al.  Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations , 2004, Physical review letters.

[37]  Arne Lüchow,et al.  On the accuracy of the fixed-node diffusion quantum Monte Carlo method , 2001 .

[38]  Trygve Helgaker,et al.  Basis-set convergence of the energy in molecular Hartree–Fock calculations , 1999 .

[39]  S. Sorella,et al.  Numerical study of the two-dimensional Heisenberg model using a Green function Monte Carlo technique with a fixed number of walkers , 1998 .

[40]  Trygve Helgaker,et al.  Basis-set convergence in correlated calculations on Ne, N2, and H2O , 1998 .

[41]  Jeppe Olsen,et al.  Surprising cases of divergent behavior in Mo/ller–Plesset perturbation theory , 1996 .

[42]  K. Schulten,et al.  Introduction to the diffusion Monte Carlo method , 1996, physics/9702023.

[43]  N. H. March,et al.  Recent progress in the field of electron correlation , 1994 .

[44]  C. Umrigar,et al.  A diffusion Monte Carlo algorithm with very small time-step errors , 1993 .

[45]  H. G. Petersen,et al.  Error estimates on averages of correlated data , 1989 .

[46]  Peter J. Knowles,et al.  A new determinant-based full configuration interaction method , 1984 .

[47]  J. A. Barker.  A quantum‐statistical Monte Carlo method; path integrals with boundary conditions , 1979 .

[48]  A. J. Walker.  New fast method for generating discrete random numbers with arbitrary frequency distributions , 1974 .

[49]  J. P. Malrieu,et al.  Iterative perturbation calculations of ground and excited state energies from multiconfigurational zeroth‐order wavefunctions , 1973 .

[50]  K. Chung.  Markov Chains with Stationary Transition Probabilities , 1961 .

[51]  A. Alavi.  9 Introduction to Full Configuration Interaction Quantum Monte Carlo with Applications to the Hubbard model , 2016 .

[52]  R. Needs,et al.  Quantum Monte Carlo simulations of solids , 2001 .

[53]  A. Sokal.  Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms , 1997 .

[54]  K. Fiedler,et al.  Monte Carlo Methods in Ab Initio Quantum Chemistry , 1995 .

[55]  K. Chung.  Further limit theorems , 1960 .