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Lecture notes in computer science

Automated Operation Minimization of Tensor Contraction Expressions in Electronic Structure Calculations

Albert Hartono, Alexander Sibiryakov, Marcel Nooijen, Gerald Baumgartner, David E. Bernholdt, So Hirata, Chi‐Chung Lam, Russell M. Pitzer, Jagannathan Ramanujam, Ponnuswamy Sadayappan

Published Jan 1, 2005• 33 citations
Publisher
DOI
10.1007/11428831_20

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We have developed a symbolic manipulation program and program generator (tensor contraction engine or TCE) that abstracts and automates the time-consuming, error-prone processes of deriving the working equations of a well-defined model of second-quantized many-electron theories and synthesizing efficient parallel computer programs on the basis of these equations. Provided an ansatz of a many-electron theory model, TCE performs valid contractions of creation and annihilation operators according to Wick's theorem, consolidates identical terms, and reduces the expressions into the form of multiple tensor contractions acted upon by permutation operators. It subsequently determines the binary contraction order for each multiple tensor contraction with the minimal operation and memory cost, factorizes common binary contractions (defines intermediate tensors), and identifies reusable intermediates. The resulting ordered list of binary tensor contractions, additions, and index permutations is translated into an optimized program that is combined with the NWCHEM and UTCHEM computational chemistry software packages. The programs synthesized by TCE take advantage of spin symmetry (within the spin−orbital formalisms), real Abelian point-group symmetry, and index permutation symmetry at every stage of the calculations to minimize the number of arithmetic operations and storage requirement, adjust the peak local memory usage by index-range tiling, and support parallel I/O interfaces and dynamic load balancing for parallel executions. We demonstrate the utility of TCE through automatic derivation and implementation of parallel programs for a range of predictive computational methods configuration-interaction theory (CISD, CISDT, CISDTQ), generalized many-body perturbation theory [MBPT(2), MBPT(3), MBPT(4)], and coupled-cluster theory (LCCD, CCD, LCCSD, CCSD, QCISD, CCSDT, and CCSDTQ), some for the first time and discuss the performance of the implemented programs.

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A Non-antisymmetric Tensor Contraction Engine for the Automated Implementation of Spin-Adapted Coupled Cluster Approaches

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We present a symbolic manipulation algorithm for the efficient automated implementation of rigorously spin-free coupled cluster (CC) theories based on a unitary group parametrization. Due to the lack of antisymmetry of the unitary group generators under index permutations, all quantities involved in the equations are expressed in terms of non-antisymmetric tensors. Given two tensors, all possible contractions are first generated by applying Wick's theorem. Each term is then put down in the form of a non-antisymmetric Goldstone diagram by assigning its contraction topology. The subsequent simplification of the equations by summing up equivalent terms and their factorization by identifying common intermediates is performed via comparison of these contraction topologies. The definition of the contraction topology is completely general for non-antisymmetric Goldstone diagrams, which enables our algorithm to deal with noncommuting excitations in the cluster operator that arises in the unitary group based CC formulation for open-shell systems. The resulting equations are implemented in a new code, in which tensor contractions are performed by successive application of matrix-matrix multiplications. Implementation of the unitary group adapted CC equations for closed-shell systems and for the simplest open-shell case, i.e., doublets, is discussed, and representative calculations are presented in order to assess the efficiency of the generated codes.

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

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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.

APL Computational Physics•Dec 1, 2025•2 citations
We have developed a symbolic manipulation program and program generator (tensor contraction engine or TCE) that abstracts and automates the time-consuming, error-prone processes of deriving the working equations of a well-defined model of second-quantized many-electron theories and synthesizing efficient parallel computer programs on the basis of these equations. Provided an ansatz of a many-electron theory model, TCE performs valid contractions of creation and annihilation operators according to Wick's theorem, consolidates identical terms, and reduces the expressions into the form of multiple tensor contractions acted upon by permutation operators. It subsequently determines the binary contraction order for each multiple tensor contraction with the minimal operation and memory cost, factorizes common binary contractions (defines intermediate tensors), and identifies reusable intermediates. The resulting ordered list of binary tensor contractions, additions, and index permutations is translated into an optimized program that is combined with the NWCHEM and UTCHEM computational chemistry software packages. The programs synthesized by TCE take advantage of spin symmetry (within the spin−orbital formalisms), real Abelian point-group symmetry, and index permutation symmetry at every stage of the calculations to minimize the number of arithmetic operations and storage requirement, adjust the peak local memory usage by index-range tiling, and support parallel I/O interfaces and dynamic load balancing for parallel executions. We demonstrate the utility of TCE through automatic derivation and implementation of parallel programs for a range of predictive computational methods configuration-interaction theory (CISD, CISDT, CISDTQ), generalized many-body perturbation theory [MBPT(2), MBPT(3), MBPT(4)], and coupled-cluster theory (LCCD, CCD, LCCSD, CCSD, QCISD, CCSDT, and CCSDTQ), some for the first time and discuss the performance of the implemented programs.
We present a symbolic manipulation algorithm for the efficient automated implementation of rigorously spin-free coupled cluster (CC) theories based on a unitary group parametrization. Due to the lack of antisymmetry of the unitary group generators under index permutations, all quantities involved in the equations are expressed in terms of non-antisymmetric tensors. Given two tensors, all possible contractions are first generated by applying Wick's theorem. Each term is then put down in the form of a non-antisymmetric Goldstone diagram by assigning its contraction topology. The subsequent simplification of the equations by summing up equivalent terms and their factorization by identifying common intermediates is performed via comparison of these contraction topologies. The definition of the contraction topology is completely general for non-antisymmetric Goldstone diagrams, which enables our algorithm to deal with noncommuting excitations in the cluster operator that arises in the unitary group based CC formulation for open-shell systems. The resulting equations are implemented in a new code, in which tensor contractions are performed by successive application of matrix-matrix multiplications. Implementation of the unitary group adapted CC equations for closed-shell systems and for the simplest open-shell case, i.e., doublets, is discussed, and representative calculations are presented in order to assess the efficiency of the generated codes.
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.