In this paper, we propose a general analysis framework for inexact power iteration, which can be used to efficiently solve high-dimensional eigenvalue problems arising from quantum many-body problems. Under this framework, we establish the convergence theorems for several recently proposed randomized algorithms, including full configuration interaction quantum Monte Carlo and fast randomized iteration. The analysis is consistent with numerical experiments for physical systems such as the Hubbard model and small chemical molecules. We also compare the algorithms both in convergence analysis and numerical results.
In this paper, we propose a general analysis framework for inexact power iteration, which can be used to efficiently solve high-dimensional eigenvalue problems arising from quantum many-body problems. Under this framework, we establish the convergence theorems for several recently proposed randomized algorithms, including full configuration interaction quantum Monte Carlo and fast randomized iteration. The analysis is consistent with numerical experiments for physical systems such as the Hubbard model and small chemical molecules. We also compare the algorithms both in convergence analysis and numerical results.
In this paper, we propose a general analysis framework for inexact power iteration, which can be used to efficiently solve high dimensional eigenvalue problems arising from quantum many-body problems. Under the proposed framework, we establish the convergence theorems for several recently proposed randomized algorithms, including the full configuration interaction quantum Monte Carlo (FCIQMC) and the fast randomized iteration (FRI). The analysis is consistent with numerical experiments for physical systems such as Hubbard model and small chemical molecules. We also compare the algorithms both in convergence analysis and numerical results.
We introduce a family of methods for the full configuration interaction problem in quantum chemistry, based on the fast randomized iteration (FRI) framework [Lim, L.-H.; Weare, J. SIAM Rev. 2017, 59, 547; DOI: 10.1137/15M1040827 ]. These methods, which we term “FCI-FRI”, 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 FCI-FRI scheme is 11–45 times more statistically efficient than the multinomial FCI-FRI scheme, which is in turn 1.4–178 times more statistically efficient than the original FCIQMC algorithm.
We present a stable and systematically improvable quantum Monte Carlo (QMC) approach to calculating excited-state energies, which we implement using our fast randomized iteration method for the full configuration interaction problem (FCI-FRI). Unlike previous excited-state quantum Monte Carlo methods, our approach, which is based on an asymmetric variant of subspace iteration, avoids the use of dot products of random vectors and instead relies upon trial vectors to maintain orthogonality and estimate eigenvalues. By leveraging recent advances, we apply our method to calculate ground- and excited-state energies of challenging molecular systems in large active spaces, including the carbon dimer with 8 electrons in 108 orbitals (8e,108o), an oxo-Mn(salen) transition metal complex (28e,28o), ozone (18e,87o), and butadiene (22e,82o). In the majority of these test cases, our approach yields total excited-state energies that agree with those from state-of-the-art methods─including heat-bath CI, the density matrix renormalization group approach, and FCIQMC─to within sub-milliHartree accuracy. In all cases, estimated excitation energies agree to within about 0.1 eV.
We present three modifications to our recently introduced fast randomized iteration method for full configuration interaction (FCI-FRI) and investigate their effects on the method’s performance for Ne, H 2 O, and N 2 . The initiator approximation, originally developed for full configuration interaction quantum Monte Carlo, significantly reduces statistical error in FCI-FRI when few samples are used in compression operations, enabling its application to larger chemical systems. The semistochastic extension, which involves exactly preserving a fixed subset of elements in each compression, improves statistical efficiency in some cases but reduces it in others. We also developed a new approach to sampling excitations that yields consistent improvements in statistical efficiency and reductions in computational cost. We discuss possible strategies based on our findings for improving the performance of stochastic quantum chemistry methods more generally.
, 547; DOI: 10.1137/15M1040827 ]. These methods, which we term "FCI-FRI", 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 FCI-FRI scheme is 11-45 times more statistically efficient than the multinomial FCI-FRI scheme, which is in turn 1.4-178 times more statistically efficient than the original FCIQMC algorithm.
Full Configuration Interaction Quantum Monte Carlo (FCIQMC) has been effectively applied to very large configuration interaction (CI) problems and was recently adapted for use as an active space solver and combined with orbital optimization. In this work, we detail an approach within FCIQMC to allow for efficient sampling of fully internally contracted multireference perturbation theories within the same stochastic framework. Schemes are described to allow for the close control over the resolution of stochastic sampling of the effective higher-body intermediates within the active space. It is found that while complete active space second-order perturbation theory seems less amenable to a stochastic reformulation, strongly contracted N-Electron Valence second-order Perturbation Theory (NEVPT2) is far more stable, requiring a similar number of walkers to converge the sc-NEVPT2 expectation values as to converge the underlying CI problem. We demonstrate the application of the stochastic approach to the computation of sc-NEVPT2 within a (24, 24) active space in a biologically relevant system and show that small numbers of walkers are sufficient for a faithful sampling of the sc-NEVPT2 energy to chemical accuracy, despite the active space already exceeding the limits of practicality for traditional approaches. This raises prospects of an efficient stochastic solver for multireference chemical problems requiring large active spaces, with an accurate treatment of external orbitals.