The sign problem in full configuration interaction quantum Monte Carlo:   Linear and sub-linear representation regimes for the exact wave function
  We investigate the sign problem for full configuration interaction quantum Monte Carlo (FCIQMC), a stochastic algorithm for finding the ground state solution of the Schr\"odinger equation with substantially reduced computational cost compared with exact diagonalisation. We find $k$-space Hubbard models for which the solution is yielded with storage that grows sub-linearly in the size of the many-body Hilbert space, in spite of using a wave function that is simply linear combination of states. The FCIQMC algorithm is able to find this sub-linear scaling regime without bias and with only a choice of Hamiltonian basis. By means of a demonstration we solve for the energy of a 70-site half-filled system (with a space of $10^{38}$ determinants) in 250 core hours, substantially quicker than the $\sim$10$^{36}$ core hours that would be required by exact diagonalisation. This is the largest space that has been sampled in an unbiased fashion. The challenge for the recently-developed FCIQMC method is made clear: expand the sub-linear scaling regime whilst retaining exact on average accuracy. This result rationalizes the success of the initiator adaptation (i-FCIQMC) and offers clues to improve it. We argue that our results changes the landscape for development of FCIQMC and related methods. 
James J. Shepherd, Gustavo E. Scuseria, and James S. Spencer
physics.comp-ph cond-mat.mtrl-sci cond-mat.other cond-mat.str-el physics.chem-ph