The recently proposed full configuration interaction quantum Monte Carlo method allows access to essentially exact ground-state energies of systems of interacting fermions substantially larger than previously tractable without knowledge of the nodal structure of the ground-state wave function. We investigate the nature of the sign problem in this method and how its severity depends on the system studied. We explain how cancelation of the positive and negative particles sampling the wave function ensures convergence to a stochastic representation of the many-fermion ground state and accounts for the characteristic population dynamics observed in simulations.
One of the major goals of electronic structure methods is to produce accurate ground-state energies and properties of many-electron systems. 1 Quantum chemistry provides a hierarchy of ab initio methods 2 based upon Hartree-Fock, of which coupled-cluster singles and doubles with perturbative triples (CCSD(T)) is the most accurate applicable to medium-sized molecules. 3 Quantum Monte Carlo methods such as Green's function Monte Carlo 4,5 (GFMC), the closely related diffusion Monte Carlo [6][7][8] (DMC), and auxiliary-field quantum Monte Carlo 9 (AFQMC) produce accurate results via a stochastic sampling of the many-electron wave function, but none of these methods is exact: GFMC and DMC simulations of all but the smallest systems converge to the physically irrelevant many-boson ground state unless the fixed-node approximation is made; 6,7 whilst AFQMC requires the phaseless approximation 10 in order to avoid an exponential growth in noise except in certain special cases. 11 The full configuration interaction (FCI) method 12 casts the Schrödinger equation as a matrix eigenvalue problem, in which the requirement that the manyelectron wave function be anti-symmetric with respect to exchange of electrons is imposed by working in a space of Slater determinants formed from a finite basis set of single-particle wave functions. The lowest eigenvalue and eigenvector of the FCI Hamiltonian matrix give the exact ground-state energy and wave function of the system, subject only to the error due to the finite basis set. Whilst the computational cost of FCI scales factorially with system size, it nevertheless represents the holy grail of electronic structure methods.
In 2009, Booth, Thom and Alavi 13 introduced a new stochastic approach in which the nodal structure of the ground-state wave function emerges spontaneously by sampling the discrete space of Slater determinants. Their “full configuration interaction quantum Monte Carlo” (FCIQMC) method yields exact (i.e. FCI-quality) results whilst requiring a fraction of the memory of an FCI cal-culation using the same basis. The memory required by FCIQMC simulations still scales factorially with system size, but the exponent appears to be substantially smaller than for FCI simulations. Moreover, unlike the iterative diagonaliation schemes required for FCI, the FCIQMC algorithm is readily parallelizable and can run efficiently on thousands of cores. Alavi and co-workers have used FCIQMC to reproduce essentially every molecular FCI calculation ever done and have obtained ground-state energies for systems with Hilbert spaces many orders of magnitude larger than the largest FCI calculations. [13][14][15] We believe that the FCIQMC method will become increasingly important in the electronic structure community, especially if improved algorithms or substantially cheaper approximations can be found. Our motivation for exploring the behavior of the method is to provide insight into possible improvements.
An FCI calculation based on iterative diagonalization (using, for example, the Davidson method) requires the storage of at least two vectors, each of length equal to the size of the Hilbert space. An FCIQMC simulation using the same basis requires the storage of the labels of the determinants occupied by a population of stochastic walkers (which, following Anderson, 6 we call “psi-particles” or psips) scattered over the same Hilbert space. In order for FCIQMC to be more efficient than FCI, the number of psips required must be a small fraction of the size of the Hilbert space. The fraction required is system dependent and provides a measure of how “hard” it is for FCIQMC to treat that system. The hardness is surprisingly difficult to predict. FCIQMC is wildly successful for some systems, such as the neon atom, where it requires only ∼0.01% of the memory of a conventional FCI calculation. 13 Even for the nitrogen molecule, a classic example of a strongly correlated system and a tough test for quantum chemical methods, FCIQMC used only a quarter of the memory of the equivalent FCI calculation. 13 However, FCIQMC struggles to describe the methane molecule, 13 for which Hartree-Fock is a very good approximation. We show in Sec. II that FCIQMC also struggles when applied to the Hubbard model and cannot treat systems larger than existing FCI methods unless U is very small. What is it that makes a system difficult? Evidently the answer is more complicated than whether or not the system is strongly correlated.
The aim of this paper is to understand the FCIQMC algorithm better. Why are some systems more difficult to treat than others? What determines the characteristic population dynamics observed in FCIQMC simulations? How does the cancelation of positive and negative psips ensure convergence to the many-fermion ground state? How many psips are required to obtain correct results? What goes wrong if the population of psips is too small? We provide at least partial answers to all of these qu
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