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APL Computational Physics

Development of stratified stochastic tensor contraction method for applications in electronic structure theory

Chandler Martin, Arick Grootveld, Anaira Jalan, Venkata Gandikota, Arindam Chakraborty

Published Dec 1, 2025• 2 citations
Publisher

Abstract

Calculation of high-rank tensor contractions plays a central role in computational physics, quantum chemistry, and computer science. The ability to perform a tensor contraction within a given computational budget poses a challenge to feasibility and restricts the types of systems that can be investigated computationally. We present the development and implementation of the Stratified Stochastic Tensor Contraction (SSTC) method as an alternative to sequential evaluation of tensor contraction. The SSTC approach partitions the tensor indices into stratified segments and employs composite index mappings to guide efficient sampling. By leveraging intrinsic structure within tensors, SSTC achieves rapid convergence to exact results with controllable sampling errors. We demonstrate the method on representative cases, two-point Coulomb integrals, four-point kernels, and correlation energy calculations using 2nd order Møller–Plesset perturbation theory, all of which are relevant to explicitly correlated electronic structure theory. In all examples, the SSTC method provides an accurate estimate of the tensor contraction with systematically reducible errors. We present the theory of exact tensor contractions as a foundation for the stochastic approach, followed by the mathematical derivation of the SSTC framework. Key components include the construction of a composite index to enable loop compression, and the application of functional minimization techniques for variance reduction. A discussion comparing sequential and stochastic contraction strategies, analysis of the distribution of sampling points through cumulative distribution functions and correlations between sampling assignments and kernel values are presented. A description of the SSTC method’s impact on computational clock times and dependence on segment number is included. Additional considerations include sampling with vs without replacement and the computational advantages of index mapping. These results highlight the effectiveness of the SSTC method as a versatile and scalable alternative to the conventional sequential tensor contraction approach, with broad applicability to quantum chemistry, many-body physics, stochastic quantum mechanics, and tensor-based machine learning.

DOI
10.1063/5.0298770

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Calculation of high-rank tensor contractions plays a central role in computational physics, quantum chemistry, and computer science. The ability to perform a tensor contraction within a given computational budget poses a challenge to feasibility and restricts the types of systems that can be investigated computationally. We present the development and implementation of the Stratified Stochastic Tensor Contraction (SSTC) method as an alternative to sequential evaluation of tensor contraction. The SSTC approach partitions the tensor indices into stratified segments and employs composite index mappings to guide efficient sampling. By leveraging intrinsic structure within tensors, SSTC achieves rapid convergence to exact results with controllable sampling errors. We demonstrate the method on representative cases, two-point Coulomb integrals, four-point kernels, and correlation energy calculations using 2nd order Møller–Plesset perturbation theory, all of which are relevant to explicitly correlated electronic structure theory. In all examples, the SSTC method provides an accurate estimate of the tensor contraction with systematically reducible errors. We present the theory of exact tensor contractions as a foundation for the stochastic approach, followed by the mathematical derivation of the SSTC framework. Key components include the construction of a composite index to enable loop compression, and the application of functional minimization techniques for variance reduction. A discussion comparing sequential and stochastic contraction strategies, analysis of the distribution of sampling points through cumulative distribution functions and correlations between sampling assignments and kernel values are presented. A description of the SSTC method’s impact on computational clock times and dependence on segment number is included. Additional considerations include sampling with vs without replacement and the computational advantages of index mapping. These results highlight the effectiveness of the SSTC method as a versatile and scalable alternative to the conventional sequential tensor contraction approach, with broad applicability to quantum chemistry, many-body physics, stochastic quantum mechanics, and tensor-based machine learning.
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.
Second-order Moller--Plesset perturbation theory (MP2) is one of the simplest correlated wave-function methods, but its conventional O(N5)O(N^5)O(N5) cost limits its application to large systems. Stochastic tensor contraction (STC) has recently appeared as a general technique to evaluate high-order tensor contractions in quantum chemistry. Here, we apply STC to the exchange contribution of Laplace-transformed MP2, which is the source of O(N5)O(N^5)O(N5) scaling in the formulation. The resulting STC exchange algorithm has an O(N3)O(N^3)O(N3) deterministic setup cost and an O(N2)O(N^2)O(N2) stochastic cost at fixed absolute error. The estimator is unbiased and provides a way to specify the target stochastic error by estimating the number of required samples before the full calculation. We implement the algorithm using a hybrid deterministic--stochastic evaluation strategy, with grouped index sampling, to reduce the computational prefactor. Over a range of benchmark molecules containing up to ∼7000\sim 7000∼7000 basis functions, the stochastic exchange evaluation takes as little as 1/2701/2701/270 of the time of a complete DF-MP2 calculation on the same system. Within the Laplace-transformed formulation, the O(N4)O(N^4)O(N4) scaling direct contribution is thus the only significant cost. Our results further substantiate the power of STC to serve as a general tensor-contraction engine for quantum chemistry.
Tensor network contraction is a fundamental computational challenge underlying quantum many-body physics, statistical mechanics, and machine learning. Belief propagation (BP) provides an efficient approximate solution, but introduces systematic errors on graphs with loops. Here, we introduce a hybrid method that achieves accurate results by stochastically sampling loop corrections to BP and showcase our method by applying it to the two-dimensional ferromagnetic Ising model. For any pairwise Markov random field with symmetric edge potentials, our approach exploits an exact factorization of the partition function into the BP contribution and a loop correction factor summing over all valid loop configurations, weighted by edge weights derived directly from the potentials. We sample this sum using Markov chain Monte Carlo with moves that preserve the loop constraint, combined with umbrella sampling to ensure efficient exploration across all correlation strengths. Our stochastic approach provides unbiased estimates with controllable statistical error in any parameter regime.
This thesis develops stochastic many-body methodologies for the electronic structure description of chemically relevant systems with strong electron correlation and nonequilibrium dynamics. Many phenomena of interest in chemistry and materials science, including excited-state processes, charge transfer at catalytic interfaces, and plasmon-mediated photocatalysis, require an accurate treatment of interacting electrons beyond mean-field theory. However, direct solutions of the many-body Schrodinger equation remain computationally intractable for realistic systems due to the exponential growth of the configurational Hilbert space with system size. This challenge motivates the development of reduced representations and embedding strategies capable of retaining essential correlation effects at lower computational cost. The work presented here focuses on stochastic matrix compression and downfolding techniques for many-body electronic structure theory. In particular, new stochastic compression approaches are introduced for the efficient representation of many-body Hamiltonians and correlated wavefunctions within reduced active subspaces. These methods extend stochastic electronic structure techniques beyond their traditional applications in many-body perturbation theory and provide a framework for treating correlated subsystems embedded within large chemical environments. Applications of these approaches are demonstrated across several classes of problems. First, stochastic downfolding methodologies are developed for the computation of charged excitations and Dyson orbitals relevant to catalytic interfaces and molecular spectroscopy. These techniques enable the treatment of large orbital spaces while maintaining accurate descriptions of electron correlation in the active region. Second, stochastic compression methods are applied to the computation of total energies in large solvated and reactive systems, where conventional correlated approaches become prohibitively expensive. Finally, the thesis explores nonequilibrium electronic dynamics at driven catalytic interfaces, including plasmon-assisted charge transfer processes. In this context, time-dependent many-body simulations reveal that transient driving fields can dynamically modify molecular spectral structure, opening additional resonant pathways for hot-carrier injection that are absent in equilibrium descriptions. Taken together, the methods developed in this thesis provide new strategies for reducing the computational complexity of correlated electronic structure calculations while preserving the essential physics of many-body interactions. These results broaden the applicability of stochastic and embedding-based approaches to large-scale equilibrium and nonequilibrium problems in chemistry, condensed matter physics, and materials science.
We review a suite of stochastic vector computational approaches for studying the electronic structure of extended condensed matter systems. These techniques help reduce algorithmic complexity, facilitate efficient parallelization, simplify computational tasks, accelerate calculations, and diminish memory requirements. While their scope is vast, we limit our study to ground-state and finite temperature density functional theory (DFT) and second-order many-body perturbation theory. More advanced topics, such as quasiparticle (charge) and optical (neutral) excitations and higher-order processes, are covered elsewhere. We start by explaining how to use stochastic vectors in computations, characterizing the associated statistical errors. Next, we show how to estimate the electron density in DFT and discuss effective techniques to reduce statistical errors. Finally, we review the use of stochastic vectors for calculating correlation energies within the second-order Møller-Plesset perturbation theory and its finite temperature variational form. Example calculation results are presented and used to demonstrate the efficacy of the methods.