Many computational methods in ab initio quantum chemistry are formulated in terms of high-order tensor contractions, whose cost determines the size of system that can be studied. We introduce stochastic tensor contraction to perform such operations with greatly reduced cost, and present its application to the gold-standard quantum chemistry method, coupled cluster theory with up to perturbative triples. For total energy errors more stringent than chemical accuracy, we reduce the computational scaling to that of mean-field theory, while starting to approach the mean-field absolute cost, thereby challenging the existing cost-to-accuracy landscape. Benchmarks against state-of-the-art local correlation approximations further show that we achieve an order-of-magnitude improvement in both total computation time and error, with significantly reduced sensitivity to system dimensionality and electron delocalization. We conclude that stochastic tensor contraction is a powerful computational primitive to accelerate a wide range of quantum chemistry.
The objective of ab initio quantum chemistry is to predict the behavior of molecules and materials from first-principles calculation of the electronic structure. Commonly used systematically improvable methods compute the quantum wavefunction of the electrons by starting from a mean-field electron approximation, and then incorporating electron correlations through contributions that are theoretically justified from perturbation theory in their interactions. (1)(2)(3)(4)(5)(6)(7)(8)(9)(10) In computational form, such techniques share a mathematical structure of tensor contractions, i.e. the behavior of the electrons (the quantum amplitudes) and their interactions are represented by arrays of numbers (tensors), which are multiplied and summed over (contracted) to yield simple output observables, such as the energy and electron densities. The computational and memory cost of these operations determines the size and complexity of systems we can study today.
A prominent example of such a method is coupled-cluster (CC) theory (1-3), most commonly used in its variant known as CCSD(T) (11)(12)(13) (denoting singles, doubles, and perturbative triples excitations). This is regarded as the gold standard of quantum chemistry, because it provides high accuracy in many applications (3,14,15). While the mean-field starting point has an O(N 4 ) cost, where N is the number of atoms, in CCSD(T) the most complex tensor contraction has a much higher O(N 7 ) cost, limiting its direct application to small problems. Much effort has been directed to reducing this scaling (16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27), most notably through the use of the local approximation (28), which captures the observation that electron correlations can be neglected when they are widely separated. In practice, this is expressed by truncating parts of the tensors that become small when expressed in a local basis, reducing the amount of computation. However, due to these truncations, local correlation methods introduce additional systematic errors, implementation complexity, and a cost which still scales steeply as a function of the electron delocalization and dimensionality of the system.
Here we introduce an alternative approach to simplify the computation of many electron correlation theories through stochastic tensor contraction (STC). While including stochastic elements when formulating correlation theories is not new (7,(29)(30)(31)(32)(33)(34)(35)(36)(37)(38), STC introduces sampling in a different and more pervasive manner to earlier works.
The computational cost of ab initio quantum chemistry simulations is usually dominated by tensor contractions. We introduce a stochastic method to evaluate such tensor contractions with greatly reduced cost. Applying stochastic tensor contraction to the gold-standard quantum chemistry method, coupled cluster with perturbative triples, we reduce its cost scaling to that of mean-field theory (while starting to approach it in absolute cost), thereby enabling larger chemical problems to be simulated at high accuracy, and demonstrate the superiority of our approach compared to the predominant numerical approximation paradigm of local correlation. Stochastic tensor contraction thus stands as a technique to accelerate a wide range of quantum chemistry methods. The main idea is to trade the exact evaluation of contractions for unbiased statistical estimates generated by a properly designed importance sampling of the elements of the contraction. We show through detailed theoretical analysis and numerical demonstration that this leads to a drastic reduction in cost, without requiring the approximation of truncating tensors in a local basis. For the example of CCSD(T), we describe an implementation with a one-time O(N 4 ) deterministic cost to set up probability tables, followed by (for a specified statistical error in the total energy) an O(N2 ) stochastic cost for CCSD, and O(N 4 ) for (T). The scaling of this goldstandard quantum chemistry method is thus reduced to that of its mean-field starting point. To support these scalings, we benchmark the practical performance of STC-CCSD(T) on a wide set of molecules against a state-ofthe-art local correlation implementation (using domain localized pair natural orbitals (22)), where we find an order of magnitude improvement in total computation time, total energy error, and a weak sensitivity to dimensionality and electron delocalization. Our work thus introduces stochastic tensor contraction as a simple but powerful computational primitive to accelerate many quantum chemistry correlation theories, and which, in practice, can already provide leading performance today.
We first introduce the general setting of tensor contractions in quantum chemistry. The overall workflow is illustrated in Fig. 1. The standard approach to electronic correlation starts from a mean-field Hartree-Fock (HF) wavefunction, which is the ground-state of an effective one-electron Hamilton
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