Exciting Determinants in Quantum Monte Carlo: Loading the Dice with Fast, Low-Memory Weights

Fighting Noise with Noise: A Stochastic Projective Quantum Eigensolver

In the current noisy intermediate scale quantum era of quantum computation, available hardware is severely limited by both qubit count and noise levels, precluding the application of many current hybrid quantum-classical algorithms to nontrivial quantum chemistry problems. In this paper we propose applying some of the fundamental ideas of conventional Quantum Monte Carlo algorithms─stochastic sampling of both the wave function and the Hamiltonian─to quantum algorithms in order to significantly decrease quantum resource costs. In the context of an imaginary-time propagation based projective quantum eigensolver, we present a novel approach to estimating physical observables which can lead to an order of magnitude reduction in the required sampling of the quantum state to converge the ground state energy of a system relative to current state-of-the-art eigensolvers. The method can be equally applied to excited-state calculations and, combined with stochastic approximations of the system Hamiltonian, provides a promising near-term approach to Hamiltonian simulation for general chemistry on quantum devices.

Exploiting Locality in Full Configuration Interaction Quantum Monte Carlo for Fast Excitation Generation

In this paper, we propose an improved excitation generation algorithm for the full configuration interaction quantum Monte Carlo method, which is particularly effective in systems described by localized orbitals. The method is an extension of the precomputed heat-bath strategy of Holmes et al., with more effective sampling of double excitations and a novel approach for nonuniform sampling of single excitations. We demonstrate the effectiveness of the algorithm for a chain of 30 hydrogen atoms with atom-localized orbitals, a stack of benzene molecules, and an Fe(II)-porphyrin model complex, whereby we show an overall efficiency gain by a factor of two to four, as measured by variance reduction per wall-clock time.

A hybrid stochastic configuration interaction-coupled cluster approach for multireference systems

The development of multireference coupled cluster (MRCC) techniques has remained an open area of study in electronic structure theory for decades due to the inherent complexity of expressing a multiconfigurational wavefunction in the fundamentally single-reference coupled cluster framework. The recently developed multireference-coupled cluster Monte Carlo (mrCCMC) technique uses the formal simplicity of the Monte Carlo approach to Hilbert space quantum chemistry to avoid some of the complexities of conventional MRCC, but there is room for improvement in terms of accuracy and, particularly, computational cost. In this paper, we explore the potential of incorporating ideas from conventional MRCC-namely, the treatment of the strongly correlated space in a configuration interaction formalism-to the mrCCMC framework, leading to a series of methods with increasing relaxation of the reference space in the presence of external amplitudes. These techniques offer new balances of stability and cost against accuracy, as well as a means to better explore and better understand the structure of solutions to the mrCCMC equations.

Full Configuration Interaction Excited-State Energies in Large Active Spaces from Subspace Iteration with Repeated Random Sparsification

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.

Stochastic Generalized Active Space Self-Consistent Field: Theory and Application

An algorithm to perform stochastic generalized active space calculations, Stochastic-GAS, is presented, that uses the Slater determinant based FCIQMC algorithm as configuration interaction eigensolver. Stochastic-GAS allows the construction and stochastic optimization of preselected truncated configuration interaction wave functions, either to reduce the computational costs of large active space wave function optimizations, or to probe the role of specific electron correlation pathways. As for the conventional GAS procedure, the preselection of the truncated wave function is based on the selection of multiple active subspaces while imposing restrictions on the interspace excitations. Both local and cumulative minimum and maximum occupation number constraints are supported by Stochastic-GAS. The occupation number constraints are efficiently encoded in precomputed probability distributions, using the precomputed heat bath algorithm, which removes nearly all runtime overhead of GAS. This strategy effectively allows the FCIQMC dynamics to a priori exclude electronic configurations that are not allowed by GAS restrictions. Stochastic-GAS reduced density matrices are stochastically sampled, allowing orbital relaxations via Stochastic-GASSCF, and direct evaluation of properties that can be extracted from density matrices, such as the spin expectation value. Three test case applications have been chosen to demonstrate the flexibility of Stochastic-GAS: (a) the Stochastic-GASSCF [5·(6, 6)] optimization of a stack of five benzene molecules, that shows the applicability of Stochastic-GAS toward fragment-based chemical systems; (b) an uncontracted stochastic MRCISD calculation that correlates 96 electrons and 159 molecular orbitals, and uses a large (32, 34) active space reference wave function for an Fe(II)-porphyrin model system, showing how GAS can be applied to systematically recover dynamic electron correlation, and how in the specific case of the Fe(II)-porphyrin dynamic correlation further differentially stabilizes the ³E_g over the ⁵A_1g spin state; (c) the study of an Fe₄S₄ cluster's spin-ladder energetics via highly truncated stochastic-GAS [4·(5, 5)] wave functions, where we show how GAS can be applied to understand the competing spin-exchange and charge-transfer correlating mechanisms in stabilizing different spin-states.

Spin-Pure Stochastic-CASSCF via GUGA-FCIQMC Applied to Iron-Sulfur Clusters

In this work, we demonstrate how to efficiently compute the one- and two-body reduced density matrices within the spin-adapted full configuration interaction quantum Monte Carlo (FCIQMC) method, which is based on the graphical unitary group approach (GUGA). This allows us to use GUGA-FCIQMC as a spin-pure configuration interaction (CI) eigensolver within the complete active space self-consistent field (CASSCF) procedure and hence to stochastically treat active spaces far larger than conventional CI solvers while variationally relaxing orbitals for specific spin-pure states. We apply the method to investigate the spin ladder in iron-sulfur dimer and tetramer model systems. We demonstrate the importance of the orbital relaxation by comparing the Heisenberg model magnetic coupling parameters from the CASSCF procedure to those from a CI-only (CASCI) procedure based on restricted open-shell Hartree-Fock orbitals. We show that the orbital relaxation differentially stabilizes the lower-spin states, thus enlarging the coupling parameters with respect to the values predicted by ignoring orbital relaxation effects. Moreover, we find that, while CASCI results are well fit by a simple bilinear Heisenberg Hamiltonian, the CASSCF eigenvalues exhibit deviations that necessitate the inclusion of biquadratic terms in the model Hamiltonian.

Binding curve of the beryllium dimer using similarity-transformed FCIQMC: Spectroscopic accuracy with triple-zeta basis sets

We demonstrate how similarity-transformed full configuration interaction quantum Monte Carlo (FCIQMC) based on the transcorrelated Hamiltonian can be applied to make highly accurate predictions for the binding curve of the beryllium dimer, marking the first case study of a molecular system with this method. In this context, the non-Hermitian transcorrelated Hamiltonian, resulting from a similarity transformation with a Jastrow factor, serves the purpose to effectively address dynamic correlation beyond the used basis set and thus allows for obtaining energies close to the complete basis set limit from FCIQMC already with moderate basis sets and computational effort. Building on results from other explicitly correlated methods, we discuss the role of the Jastrow factor and its functional form, as well as potential sources for size consistency errors, and arrive at Jastrow forms that allow for high accuracy calculations of the vibrational spectrum of the beryllium dimer.

Stochastic perturbation theory in a limited configuration space

A general-order stochastic perturbation algorithm is obtained from the order-by-order expansion of the imaginary-time evolution of a configuration interaction wave function. A truncation of configuration space that is required for the practical treatment of the perturbative corrections, however, does not preserve size-consistency as is the case for a truncated configuration interaction. To circumvent this problem, we formulate a linked variant of stochastic perturbation theory based on the coupled-cluster ansatz. The implementation based on the linearized coupled-cluster is compared with several full configuration interaction results. We also compare the results with those obtained from deterministic coupled-cluster and many-body perturbation theories.