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arXiv:2602.15129

A CPD-enabled low-scaling environment solver in a coupled cluster based static quantum embedding theory

Karl Pierce, Muhammad Talha Aziz, Avijit Shee, Fabian M. Faulstich

Published Feb 16, 2026• 0 citations
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Abstract

We incorporate a canonical polyadic decomposition (CPD) based low-level solver as a means to accelerate the environment-level solver for the recently developed MPCC embedding framework. Using CPD, we both factorize the three dominant order-three density-fitting two-electron integral (DF TEI) tensors and develop a novel formulation that reduces the storage complexity of the low-level solver from ${O}(N^3)$ to $O(NR)$, where $R$ is the CPD rank, and the computational scaling of the most time-consuming contractions from ${O}(N^4)$ to ${O}(NR^2)$. We provide benchmarks on representative chemical environments, namely water clusters $\mathrm{(H_2O)_n}$ with $n = 1$ to $6$ and linear alkane chains $\mathrm{C_nH_{2n+2}}$ with $n = 1$ to $6$. For both test sets, using the CPD-compressed DF TEI tensors reproduces the DF reference convergence behavior of the low-level solver, the subsequent high-level step, and the fully self-consistent MPCC iterations, while introducing only small, rank-controlled shifts in absolute energies. At a fixed tolerance in the absolute MPCC energy, the CP ranks required for these tensor approximations increase linearly with system size. Chemically relevant energy differences are likewise preserved, as demonstrated for water-cluster dissociation energies and in a proof-of-concept embedding calculation of methane in a four-water cluster.

DOI
10.48550/arxiv.2602.15129

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We incorporate a canonical polyadic decomposition (CPD) based low-level solver as a means to accelerate the environment-level solver for the recently developed MPCC embedding framework. Using CPD, we both factorize the three dominant order-three density-fitting two-electron integral (DF TEI) tensors and develop a novel formulation that reduces the storage complexity of the low-level solver from O(N3){O}(N^3)O(N3) to O(NR)O(NR)O(NR), where RRR is the CPD rank, and the computational scaling of the most time-consuming contractions from O(N4){O}(N^4)O(N4) to O(NR2){O}(NR^2)O(NR2). We provide benchmarks on representative chemical environments, namely water clusters (H2O)n\mathrm{(H_2O)_n}(H2​O)n​ with n=1n = 1n=1 to 666 and linear alkane chains CnH2n+2\mathrm{C_nH_{2n+2}}Cn​H2n+2​ with n=1n = 1n=1 to 666. For both test sets, using the CPD-compressed DF TEI tensors reproduces the DF reference convergence behavior of the low-level solver, the subsequent high-level step, and the fully self-consistent MPCC iterations, while introducing only small, rank-controlled shifts in absolute energies. At a fixed tolerance in the absolute MPCC energy, the CP ranks required for these tensor approximations increase linearly with system size. Chemically relevant energy differences are likewise preserved, as demonstrated for water-cluster dissociation energies and in a proof-of-concept embedding calculation of methane in a four-water cluster.
We incorporate a canonical polyadic decomposition (CPD)-based low-level solver as a means to accelerate the environment-level solver for the recently developed Møller-Plesset Coupled-Cluster (MPCC) embedding framework. Using CPD, we both factorize the three dominant order-three density-fitting two-electron integral (DF TEI) tensors and develop a novel formulation that reduces the storage complexity of the low-level solver from O(N3) to O(NR), where R is the CPD rank, and the computational scaling of the most time-consuming contractions from O(N4) to O(NR2). We provide benchmarks on representative chemical environments, namely water clusters (H2O)n with n = 1 to 6 and linear alkane chains CnH2n+2 with n = 1 to 6. For both test sets, using the CPD-compressed DF TEI tensors reproduces the DF reference convergence behavior of the low-level solver, the subsequent high-level step, and the fully self-consistent MPCC iterations, while introducing only small, rank-controlled shifts in absolute energies. At a fixed tolerance in the absolute MPCC energy, the CP ranks required for these tensor approximations increase linearly with system size. Chemically relevant energy differences are likewise preserved, as demonstrated for water-cluster dissociation energies and in a proof-of-concept embedding calculation of methane in a four-water cluster.
We investigate a novel approach to approximate tensor-network contraction via the exact, matrix-free decomposition of full tensor-networks. We study this method as a means to eliminate the propagation of error in the approximation of tensor-networks. Importantly, this decomposition-based approach is generic, i.e. it does not depend on a specific tensor-network, the tensor index (physical) ordering, or the choice of tensor decomposition. Careful consideration should be made to determine the best decomposition strategy. Furthermore, this method does not rely on robust cancellation of errors (i.e. the Taylor expansion). As a means to study the effectiveness of the approach, we replace the exact contraction of the particle particle ladder (PPL) tensor diagram in the popular coupled-cluster with single and double excitation (CCSD) method with a low-rank tensor decomposition, namely the canonical polyadic decomposition (CPD). With this approach, we replace an O(N6)\mathcal{O}(N^6)O(N6) tensor contractions with a potentially reduced-scaling O(N4R)\mathcal{O}(N^4R)O(N4R) optimization problem, where RRR is the CP rank, and we reduce the computational storage of the PPL tensor from O(N4)\mathcal{O}(N^4)O(N4) to O(NR)\mathcal{O}(NR)O(NR), although we do not take advantage of this compression in this study. To minimize the cost of the CPD optimization, we utilize the iterative structure of CCSD to efficiently initialize the CPD optimization. We show that accurate chemically-relevant energy values can be computed with an error of less than 1 kcal/mol using a relatively low CP rank.
An improved algorithm to implement our previously proposed static quantum embedding method MPCC (J. Chem. Phys. 161, 164107 (2024)) is presented. It uses the density-fitting (DF) approximation for the electron-repulsion integrals (ERIs) across all layers of the embedding method: the low-level method, the screened interaction, and the high-level method. Additionally, an approximate factorized solution of the Sylvester equation is employed based on the Laplace transformation for the low-level method. To build the screened interaction, we have explored an approximate construction of it based on the distinguishable cluster approximation (DCA) by Kats and Manby (J. Chem. Phys. 141, 061101 (2014)), which amounts to neglecting certain higher-order exchange and ladder diagrams. Combining all these techniques, we have obtained an O(N4)\mathcal{O}(N^4)O(N4) algorithm for the low-level method; a non-iterative NfragO(N4)N_{\rm frag} \mathcal{O}(N^4)Nfrag​O(N4) algorithm for the screened interaction; and an iterative O(Nfrag6)\mathcal{O}(N_{\rm frag}^6)O(Nfrag6​) algorithm for the fragment solver, where NfragN_{\rm frag}Nfrag​ is the number of fragment orbitals and NNN is the total number of orbitals. The resulting low-scaling implementation of the MPCC method is applied to trans polyacetylenes, that is, C2n_{2n}2n​H2n+2_{2n+2}2n+2​ molecules for chain lengths n=1−8n=1-8n=1−8, and the timing and accuracy of the method are compared to the original unfactorized implementation. Moreover, the approximations are further validated by applying the method to the potential energy surface of the \ce{O3} molecule. Finally, we apply the method to the selected molecules in the S22 dataset for which the performance of the MP2 method is known to be poor. The results demonstrate that the DF-based implementation of the MPCC method provides a significant reduction in computational cost while maintaining accuracy comparable to that of the original implementation.
We investigate the efficient combination of the canonical polyadic decomposition (CPD) and tensor hyper-contraction (THC) approaches. We first present a novel low-cost CPD solver that leverages a precomputed THC factorization of an order-4 tensor to efficiently optimize the order-4 CPD with O ( N R 2 ) scaling. With the matrix-free THC-based optimization strategy in hand, we can efficiently generate CPD factorizations of the order-4 two-electron integral tensors and develop novel electronic structure methods that take advantage of both the THC and CPD approximations. Next, we investigate the application of a combined CPD and THC approximation of the Laplace transform (LT) second-order Mo̷ller-Plesset (MP2) method. We exploit the ability to switch efficiently between the THC and CPD factorizations of the two-electron integrals to reduce the computational complexity of the LT MP2 method while preserving the accuracy of the approach. Furthermore, we take advantage of the robust fitting approximation to eliminate the leading-order error in the CPD approximated tensor networks. Finally, we show that modest values of THC and CPD rank preserve the accuracy of the LT MP2 method and that this CPD + THC LT MP2 strategy realizes a performance advantage over canonical LT MP2 in both computational wall-times and memory resource requirements.
We investigate the efficient combination of the canonical polyadic decomposition (CPD) and tensor hyper-contraction (THC) approaches. We first present a novel low-cost CPD solver which leverages a precomputed THC factorization of an order-444 tensor to efficiently optimize the order-444 CPD with O(NR2)\mathcal{O}(NR^2)O(NR2) scaling. With the matrix-free THC-based optimization strategy in hand we can: efficiently generate CPD factorizations of the order-4 two-electron integral tensors; and develop novel electronic structure methods which take advantage of both the THC and CPD approximations. Next, we investigate the application of a combined CPD and THC approximation of the Laplace transform (LT) second-order Møller-Plesset (MP2) method. We exploit the ability to switch efficiently between the THC and CPD factorizations of the two electron integrals to reduce the computational complexity of the LT MP2 method while preserving the accuracy of the approach. Furthermore we take advantage of the robust fitting approximation to eliminate leading order error in the CPD approximated tensor networks. Finally, we show that modest values of THC and CPD rank preserve the accuracy of the LT MP2 method and that this CPD+THC LT MP2 strategy realizes a performance advantage over canonical LT MP2 in both computational wall-times and memory resource requirements.
In a previous publication, we have discussed the usage of tensor decomposition in the canonical polyadic (CP) tensor format for electronic structure methods. There, we focused on two-electron integrals and second order Møller-Plesset perturbation theory (MP2). In this work, we discuss the CP format for Coupled Cluster (CC) theory and present a pilot implementation for the Coupled Cluster Doubles method. We discuss the iterative solution of the CC amplitude equations using tensors in CP representation and present a tensor contraction scheme that minimizes the effort necessary for the rank reductions during the iterations. Furthermore, several details concerning the reduction of complexity of the algorithm, convergence of the CC iterations, truncation errors, and the choice of threshold for chemical accuracy are discussed.