Abstract
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.
Highlights
Exact methods for solving the Schrodinger equation are used at the forefront of understanding in condensed matter physics [1,2,3,4] and in molecular quantum chemistry [5,6,7]
Quantum Monte Carlo (QMC) techniques attempting to determine these parameters are hindered by their values being either positive or negative, causing more pronounced variability than a set of parameters with a single sign
We investigate the sign problem of a recently developed QMC method developed for use in finite molecular basis sets: full configuration interaction QMC (FCIQMC) [5]
Summary
Exact methods for solving the Schrodinger equation are used at the forefront of understanding in condensed matter physics [1,2,3,4] and in molecular quantum chemistry [5,6,7]. This is the direct (ground-state) QMC analog of exact diagonalization, finding the exact lowest-energy solution for a finite Hilbert space with an exponential number of states using a walker-based algorithm, where the Hilbert space is a set of Slater determinants and grows exponentially with the number of fermions Since it does not impose the signs of the wave function in advance, this method does in general have a sign problem [13], and the cost of the storage of the exact-on-average wave function has been shown to scale linearly in the size of the Hilbert space for a series of atomic systems [14,15,16]. In light of this, QMC for the FCI problem could receive routine use for the treatment of correlated electrons in many more realistic contexts
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