Motivated by the recent success of quantum neural states, we present a variational implementation of the coupled-cluster method suitable for quantum Monte Carlo simulations. We introduce a Young tableaux representation of the wave-function coefficients and their derivatives that substantially reduces the computational complexity of the variational optimization. While the algorithm is generic, we illustrate its implementation with an application to the Heisenberg model in one and two dimensions, using the block-correlated coupled-cluster Ansatz with doubles and triples. Convergence is greatly improved by means of importance sampling based on a fictitious temperature. The resulting approach achieves competitive accuracy while providing a scalable framework for variational coupled-cluster calculations.
Motivated by the recent success of quantum neural states, we present a variational implementation of the coupled-cluster method suitable for quantum Monte Carlo simulations. We introduce a Young tableaux representation of the wave-function coefficients and their derivatives that substantially reduces the computational complexity of the variational optimization. While the algorithm is generic, we illustrate its implementation with an application to the Heisenberg model in one and two dimensions, using the block-correlated coupled-cluster Ansatz with doubles and triples. Convergence is greatly improved by means of importance sampling based on a fictitious temperature. The resulting approach achieves competitive accuracy while providing a scalable framework for variational coupled-cluster calculations.
We introduce a many-body method that combines two powerful many-body techniques, viz., quantum Monte Carlo and coupled cluster theory. Coupled cluster wave functions are introduced as importance functions in a Monte Carlo method designed for the configuration interaction framework to provide rigorous upper bounds to the ground-state energy. We benchmark our method on the homogeneous electron gas in momentum space. The importance function used is the coupled cluster doubles wave function. We show that the computational resources required in our method scale polynomially with system size. Our energy upper bounds are in very good agreement with previous calculations of similar accuracy, and they can be systematically improved by including higher order excitations in the coupled cluster wave function.
Abstract Variational optimization of neural-network representations of quantum states has been successfully applied to solve interacting fermionic problems. Despite rapid developments, significant scalability challenges arise when considering molecules of large scale, which correspond to non-locally interacting quantum spin Hamiltonians consisting of sums of thousands or even millions of Pauli operators. In this work, we introduce scalable parallelization strategies to improve neural-network-based variational quantum Monte Carlo calculations for ab-initio quantum chemistry applications. We establish GPU-supported local energy parallelism to compute the optimization objective for Hamiltonians of potentially complex molecules. Using autoregressive sampling techniques, we demonstrate systematic improvement in wall-clock timings required to achieve coupled cluster with up to double excitations baseline target energies. The performance is further enhanced by accommodating the structure of resultant spin Hamiltonians into the autoregressive sampling ordering. The algorithm achieves promising performance in comparison with the classical approximate methods and exhibits both running time and scalability advantages over existing neural-network based methods.
Understanding strongly correlated quantum many-body systems requires dealing with a largeconfiguration space comprised of a set of all possible measurable configurations. The total number of configurations grows exponentially with the size of the system, which makes exact numerical calculations impossible for even moderately sized systems. Since the advent of high-temperature superconductivity and the subsequent interest in strongly correlated phases of matter, progress in the field has been marked by ingenuity of the field to overcome these computational limitations. The first techniques to emerge were inspired by quantum chemistry and based on exact diagonalization[1], which is limited to small clusters, and different variants of quantum Monte Carlo[2], that suffers from the sign problem in fermionic and frustrated systems. Circa 1992, the density matrix renormalization group (DMRG) [3, 4, 5, 6, 7] became one of the major breakthroughs of the past few decades. Despite the success of DMRG for one-dimensional(1D) and quasi-one-dimensional geometries, extensions to actual two-dimensional systems remain challenging and applications are constrained to long cylinders and strips. One promising alternative is to combine one the oldest methods, variational wave functions (VWFs), with neural networks, which have been used to recently with great success in machine learning. The focus of this dissertation is on developing novel variational Monte Carlo algorithms to make neural network variational wave functions applicable to many problems in quantum many-body physics. These algorithms draw from existing variational wave-function, tensor network, and machine learning methods. We calculate dynamical spectral functions using the vector correction and Chebyshev expansion method. We do finite temperature calculations with minimally entangled thermal states. Finally, we do ground state calculations for the frustrated J1−J2 Heisenberg model. We benchmark these results using restricted Boltzmann machines as the neural network for the the variational wave-functions.--Author's abstract
Ab initio quantum Monte Carlo (QMC) methods are state-of-the-art electronic structure calculations based on highly parallelizable stochastic frameworks for accurate solutions of the many-body Schrödinger equation, suitable for modern many-core supercomputer architectures. Despite its potential, one of the major drawbacks that still hinders QMC applications, especially when targeting dynamical properties of large systems or extensive datasets, is the lack of an affordable method to compute atomic forces that are consistent with the corresponding potential energy surfaces (PESs), also known as unbiased atomic forces. Recently, one of the authors in the present paper proposed a way to obtain unbiased forces with the Jastrow-correlated Slater determinant Ansatz, where the determinant part is frozen to the values obtained by a mean-field method, such as density functional theory [K. Nakano, M. Casula, and G. Tenti, Phys. Rev. B 109, 205151 (2024)]. However, the proposed method has a significant drawback for its applications: for a system with N nuclei, one requires 6N additional density functional theory (DFT) calculations to get unbiased forces, which is not negligible as the system size increases. This paper presents a way to replace the 6N DFT calculations with a single coupled-perturbed Kohn-Sham calculation, following the so-called Lagrangian technique established in quantum chemistry. This improves the computational cost and scalability of the method. We also demonstrate that the developed unbiased variational Monte Carlo (VMC) force calculation improves not only the consistency with PESs but also its accuracy, by investigating three molecules from the rMD17 benchmark set, and comparing the unbiased VMC forces with those obtained by the coupled-cluster singles and doubles with perturbative triples [CCSD(T)] calculations. We found that the bare VMC forces are biased from the CCSD(T) ones, while the unbiased ones give values closer to those of the CCSD(T) ones. Our benchmark test also reveals that the unbiased VMC forces yield very consistent values with hybrid and meta generalized gradient approximations (e.g., ωB97X-D3BJ and ωB97M-D3BJ), but do not necessarily yield values that are very close to those of CCSD(T). Our finding paves the way to generate machine learning interatomic potentials based on VMC forces more efficiently and accurately.
The general formalism of the block-correlated coupled cluster (BCCC) method, an alternative multireference coupled cluster method for calculating the ground-state electronic structures of molecular systems, has been presented. The BCCC theory is constructed in terms of a complete set of many-electron states of individual blocks, assumed that the whole system could be partitioned into a set of blocks. The reference state in the BCCC is selected as a tensor product of the most important many-electron state of each system block. By truncating the cluster operator to a certain n-block correlation level, an approximate but size-extensive BCCC method, denoted as BCCCn, is defined. For reducing the computational effort but without much loss of accuracy, the reduced density matrix is introduced to generate an optimal subset of many-electron states for each block. I have implemented the BCCCn (n=2,3) methods within the S=1/2 Heisenberg Hamiltonian, and applied them to calculate the ground-state energies of one-dimensional spin chains and quasi-one-dimensional two-leg spin ladders. The calculated results show that with the appropriate partition of the studied systems the BCCC3 method can yield quite satisfactory ground-state energies for these spin systems.
Abstract Neural-network quantum states (NQSs) offer a powerful and expressive ansatz for representing quantum many-body wave functions. However, their training via Variational Monte Carlo (VMC) methods remains challenging. It is well known that some scenarios—such as sharply peaked wave functions emerging in quantum chemistry—lead to high-variance gradient estimators hindering the effectiveness of variational optimizations. In this work we investigate a systematic strategy to tackle those sampling issues by means of adaptively tuned importance sampling. Our approach is explicitly designed to target the gradient estimator instead of the loss function, and be computationally inexpensive. We benchmarked our approach across the ground-state search of a wide variety of hamiltonians, including frustrated spin systems and ab-initio quantum chemistry. Overall, our approach can reduce the computational cost of vanilla VMC considerably, up to a factor of 100× when targeting highly peaked quantum chemistry wavefunctions.