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. 2024 Feb 5;20(3):1185–1192. doi: 10.1021/acs.jctc.3c01190

Distributed Implementation of Full Configuration Interaction for One Trillion Determinants

Hong Gao 1,*, Satoshi Imamura 1, Akihiko Kasagi 1, Eiji Yoshida 1
PMCID: PMC10867839  PMID: 38314701

Abstract

graphic file with name ct3c01190_0009.jpg

Full configuration interaction (FCI) can provide an exact molecular ground-state energy within a given basis set and serve as a benchmark for approximate methods in quantum chemical calculations, including the emerging variational quantum eigensolver. However, its exponential computational and memory requirements easily exceed the capability of a single server and limit its applicability to large molecules. In this paper, we present a distributed FCI implementation employing a hybrid parallelization scheme with multithreading and multiprocessing to expand FCI’s applicability. We optimize this scheme to minimize the bottlenecks arising from interprocess communications and interthread data management. Our implementation achieves higher scalability than the naive combination of prior works and successfully calculates the exact energy of C3H8/STO-3G with 1.31 trillion determinants, which is the largest FCI calculation to the best of our knowledge. Furthermore, we provide a comprehensive list of FCI results with 136 combinations of molecules and basis sets for future evaluation and development of approximate methods.

1. Introduction

Ground-state energy calculation is a basic task in quantum chemistry because it provides insights into molecular characteristics, such as the stability, chemical reactions, electronic structure, and total energy of a quantum system. It has been widely applied in materials science, condensed matter physics, biochemistry, drug discovery, and so forth.1−4

Various methods exist for calculating ground-state energy: Hartree–Fock, density functional theory, Møller–Plesset perturbation theory, coupled cluster, and so forth.5−8 In particular, the full configuration interaction (FCI) method serves as an accuracy benchmark to evaluate approximate methods because it theoretically provides the exact solution for the Schrödinger equation within a given basis set by accounting for all possible electronic configurations. However, its applicability is limited to small molecules and basis sets due to the exponential computational and memory complexity according to the number of electrons and molecular orbitals (MOs).9,10

Since the 1980s, a lot of effort has been made to expand the FCI’s applicability. Handy et al. decomposed the two-electron part of the Hamiltonian as a sum over intermediate singly excited states.11−14 Olsen et al. improved this theory and successfully calculated over one billion Slater determinants, as shown by the left red point in Figure 1.15

Figure 1.

Figure 1

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Expansion of FCI applicability.

In the early 2000s, the development of supercomputers encouraged distributed FCI implementations using message passing interface (MPI) and increased the number of determinants to ten billion, as shown by the middle and right red points in Figure 1.16−18 However, distributed implementations suffer from longer interprocess communication times, as the number of processes increases. For instance, Ansaloni et al. observed that with 128 processes, the communication time reached 26,000 s, while the calculation time was 500 s.16 Unfortunately, energy results that can be used to evaluate the accuracy of appropriate methods were not provided in these works.

Since 2005, the FCI research has shifted toward the development of two approximate CI methods: selected CI and FCI quantum Monte Carlo (FCIQMC). The selected CI methods select only important portions of Slater determinants,19−32 while the FCIQMC methods sample the FCI wave function using a stochastic approach.31−38 Although these methods can be applied to larger molecules compared with the exact FCI, their accuracy depends on the number of selected configurations.

In 2017, a multithreaded FCI implementation with open multi-processing (OpenMP) was presented in an open-source Python-based Simulations of Chemistry Framework (PySCF) to fully utilize the potential of multicore processors.39 However, since there is no distributed FCI implementation in PySCF, its applicability is limited to 18 electrons with 18 orbitals, which corresponds to one billion Slater determinants.40 Although there is a MPI implementation of PySCF called mpi4pyscf,41 FCI is not included in it.

More recently, the development of emerging quantum chemical calculations (QCC) methods has further amplified the significance of large-scale exact FCI. The advent of noisy intermediate-scale quantum computers has driven the development of quantum computing-based methods such as the variational quantum eigensolver (VQE) and quantum Monte Carlo (QMC).42−52 These methods have demonstrated superior accuracy compared to traditional QCC methods, such as coupled-cluster singles and doubles with perturbative triples [CCSD(T)] in some cases. However, the exact FCI energies necessary for their accuracy evaluation are lacking, even for relatively large molecules such as C3H6/STO-3G.53

In this paper, to calculate the exact ground-state energies of larger molecules than prior works, we present a distributed FCI implementation using a hybrid MPI-OpenMP scheme based on the OpenMP implementation in PySCF. We find that a naive hybrid MPI-OpenMP implementation based on prior works17,39 suffers from significant MPI communication time and processing time required for handling intermediate buffers when the number of processes is large. To address this issue, we optimize MPI communication and propose thread-safe cyclic data management to eliminate the intermediate buffers.

Compared to the naive distributed implementation, our optimized implementation achieves a 55% time reduction for BH3/STO-3G. We also confirm that our implementation scales better to the number of processes than the naive implementation. As a result, we successfully calculated the exact ground-state energy of C3H8/STO-3G, comprising 1.3 trillion (1012) determinants, by running 512 processes on 256 servers in 113.6 h. To the best of our knowledge, this is the largest-scale FCI calculation, as shown by the star point in Figure 1. Table 1 also shows the comprehensive comparison between our work and prior works. Unlike prior works that limited the active space, we calculate the exact ground-state energies of 136 combinations of molecules and basis sets in the full active space and present all of the results in the Supporting Information.

Table 1. Comprehensive Comparison between Prior Works and Our Work.

  SC ’0518 PySCF ’2040 our work
molecule/basis set C2/cc-pVTZ(+1s,1p) - C3H8/STO-3G
active spacea (8 elecs, 66 orbs) (18 elecs, 18 orbs) (26 elecs, 23 orbs)
full space NO YES YES
# of determinants 6.5 × 1010 2.4 × 109 1.3 × 1012
# of servers 108 1 256
energy results not provided not provided 136 results provided
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a

Active space is defined with the number of electrons and the number of orbitals.

We believe that this work would contribute to the accuracy evaluation of approximate QCC methods for large molecules that have not ever been evaluated exactly. For instance, Table 2 shows an example to evaluate the accuracy of CCSD(T), VQE, and QMC53 for C3H6/STO-3G. While it was previously impossible to determine the most accurate method without the exact FCI energy, our distributed implementation now enables this by executing FCI on 24 servers in 6 h (see Section 3 for the experimental setup). We now know that the QMC method achieves an accuracy higher than that of CCSD(T) and VQE.

Table 2. Example to Evaluate the Accuracy of Approximate Methods for C3H6/STO-3g Using Our Distributed FCI Implementation.

method energy [Hartree] error from FCI [Hartree]
distributed FCI –115.887177 -
CCSD(T) –115.886414 0.000763
VQEa –113.832597 2.054580
QMC53 –115.886571 0.000606
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a

Executed with a noise-less AerSimulator in Qiskit54 in an active space of 8 electronics and 8 orbitals.

The organization of this article is as follows. Section 2 provides background knowledge on FCI and details of our hybrid MPI-OpenMP implementation. Section 3 presents the evaluation results of our implementation. Finally, Section 4 concludes this work. A comprehensive list of energy results can be found in the Supporting Information.

2. Methods

2.1. FCI Theory

The goal of FCI is to find an exact solution to the electronic Schrödinger equation

2.1. 1

where Ĥ represents the Hamiltonian operator and Ψ stands for a wave function. Knowles and Handy proposed an algorithm that represents Ψ with a linear combination of Slater determinants ΦI.14 In their algorithm, Slater determinants are constructed by combining α-strings and β-strings, which represent the arrangement of α-spin (spin-up) and β-spin (spin-down) electrons in MOs, respectively. The α- and β-strings are bit-strings, where “1” indicates an occupied MO and “0” signifies an empty MO. The total number of Slater determinants Ndet is calculated as follows (assuming a closed-shell singlet state)

2.1. 2

where No represents the number of MOs and Ne = Nα = Nβ represents the number of electrons with α-spin, which is half of the total number of electrons.

The exact solution of eq 1 can be calculated by diagonalizing a Hamiltonian matrix H, where each element is calculated as Inline graphic. However, the direct diagonalization of the entire H is considered impractical because the size of H is the same as Ndet, which increases exponentially with the number of electrons and MOs. Thus, variational principles such as the Davidson diagonalization method are typically employed for FCI.55 They operate on subspaces of H while progressively refining the approximation of the eigenvalues and eigenvectors. Specifically, H is projected onto a set of subspace vectors, and a subspace matrix is iteratively expanded by appending components calculated with subspace vectors. When the number of subspace vectors exceeds a predetermined threshold, the subspace matrix is reset to limit the memory requirement.

The most time-consuming step in each Davidson iteration is the calculation of a subspace vector σ with a Hamiltonian matrix element HIJ and CI coefficient vector cJ as follows

2.1. 3

The algorithm shows the pseudocode of this calculation.12,56 It includes five steps: forming the D matrix with α-contributions (lines 1–8) and β-contributions (lines 9–16), calculating the E matrix through matrix–matrix multiplication (line 17), and constructing σ with α-contributions (lines 18–25) and β-contributions (lines 26–33). α′- and β′-strings represent α- and β-strings that have at most one occupation different from that of α- and β-strings, respectively. Given an α-string, the corresponding α′-strings, indexes ij (i.e., link[α, α′]), and one-electronic coupling coefficients γ[α,α′] are stored locally. For instance, such information for both α- and β-strings is preserved in a lookup table (LT) in the FCI implementation of PySCF. Therefore, the α- and β-contributions to D or σ can be calculated by referring to the LT, multiplying γ with the specified element of C or E, and subsequently adding the result to the specified element of D or σ. Matrix–matrix multiplication to calculate E can be executed with well-known libraries, such as the BLAS DGEMM library.2.1.

2.2. Optimized Hybrid Parallel Implementation

We employ a hybrid parallelization scheme combining multiprocessing with MPI and multithreading with OpenMP to fully utilize the potential of multiple servers equipped with multicore processors. In the following subsections, we first present our optimized OpenMP implementation and then describe our optimized hybrid MPI-OpenMP implementation. In addition, we introduce a restart function for long-time executions.

2.2.1. Optimized OpenMP Implementation

We implement a multithreaded FCI code with OpenMP based on the PySCF implementation, as shown in the algorithm and Figure 2. The columns (β-strings) of C and σ are partitioned into fixed-size βchunks to limit the amount of interprocess communication at a time (see Section 2.2.2 in detail). For efficient multithreaded processing, we apply a thread-safe cyclic data assignment method that further divides a βchunk into Nthreads blocks and calculates the β-contributions in each block (βchunkbid) in parallel with multiple threads. The blocks processed by multiple threads are exchanged across subloops to enable different threads to safely handle different blocks in turn. This optimization is a major improvement from the PySCF implementation, where multiple threads concurrently process the same βchunk using thread-local intermediate buffers, and the elements of σ are calculated by adding these buffers. The PySCF implementation suffers from a significant time overhead to reset and add the intermediate buffers, whereas our thread-safe cyclic data assignment eliminates them and enables each thread to directly store the elements of σ.2.2.1.

Figure 2.

Figure 2

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Our proposed OpenMP implementation of FCI. This is an example with two threads.

On the other hand, the rows (α-strings) of C and σ are distributed to multiple threads and processed in parallel. This is the same scheme as the PySCF implementation. Since there is no data dependency between different α-strings, this scheme can also be applied to our MPI parallelization (see Section 2.2.2 in detail).

2.2.2. Optimized Hybrid MPI-OpenMP Implementation

For molecules whose computational and memory requirements exceed the capability of a single server, we distribute C and σ to multiple processes and run them on multiple servers. In such a distributed implementation, interprocess communication with MPI is a typical performance bottleneck. In fact, prior distributed implementations suffered from a significant communication time as the number of servers increased.16,17

Figure 3 illustrates the overview of our hybrid MPI-OpenMP implementation. We apply a static load balancing scheme presented by Gan and Harrison, as shown in Figure 4a.17 It evenly distributes C into multiple processes by row (e.g., CP0 and CP1 for two processes in this figure). If there are remaining rows, they are further distributed evenly among processes so that the load difference between processes is at most one row. The elements of a βchunk are gathered from all processes, and then each process computes each fragment locally with multiple threads, as mentioned in Section 2.2.1. Finally, the elements of σ are calculated by aggregating the results from all processes.

Figure 3.

Figure 3

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Our optimized hybrid MPI-OpenMP implementation. This is an example with two processes, each of which has two threads.

Figure 4.

Figure 4

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Examples of (a) static load balancing, (b) Allgatherv communication, and (c) Allgather communication.

To gather the elements of a βchunk from all processes, we use the Allgather function in MPI. Since a prior MPI implementation17 used the Allgatherv function, we show the difference between them in Figure 4. Allgatherv can communicate variable-size data by including metadata containing the data type and length, which simplifies MPI programming. However, it must communicate the metadata in addition to the data itself and can cause an imbalance of communication among multiple processes. In an imbalanced case, processes handling less data must wait for those handling more data at a barrier point. In contrast, Allgather requires no metadata by communicating fixed-size data and can avoid a communication imbalance. Note that data must be aligned by padding with zeros in an unnecessary region. The combination of static load balancing in the FCI code and Allgather can minimize the time overhead of MPI communication.

To aggregate the results from all processes for calculating the elements of σ, we use the Alltoall function in MPI. In contrast, the prior MPI implementation17 combined the Reduce and Scatter functions for this purpose. Figure 5 compares the amounts of communication traffic between them. Supposing each process computes a data block whose size is d, the prior implementation needs to gather data blocks from all processes to a single process and add them with Reduce, followed by Scatter, which distributes the results among all processes. Thus, the total amount of communication traffic is (d × Nprocs2 + d × Nprocs). In contrast, our MPI implementation reduces it to d × Nprocs2 with Alltoall, which only exchanges data blocks each process does not have. Although we need to add all data blocks after Alltoall, it can be performed locally within each process without interprocess communications.

Figure 5.

Figure 5

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Comparison between Reduce + Scatter and Alltoall.

Moreover, we optimize the size of each block in a βchunk (i.e., the size of βchunkbid) that is processed by each thread. Figure 6 shows the execution time of one Davidson iteration with different sizes of βchunkbid for C2N2/STO-3G with four processes running on two nodes. We observe that the time for accessing LT and performing DGEMM is minimized when the βchunkbid size is 64 or 112. Note that the default size in PySCF is 112. On the other hand, the communication time is minimized when the βchunkbid size is 64. Thus, we set the βchunkbid size to 64 to minimize the time of each Davidson iteration.

Figure 6.

Figure 6

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Execution time of one Davidson iteration with the different size of βchunkbid for C2N2/STO-3G with four processes running on two nodes.

2.2.3. Restart Function

For large-scale FCI calculations, FCI can take a few days and abort due to an unexpected system shutdown. Therefore, we implement a restart function based on the out-of-core execution of the PySCF implementation, where the subspaces of C and σ are dumped to files. In our FCI implementation, each process saves the intermediate result calculated with the dumped subspaces in the latest Davidson iteration into files specified by a target molecule, a basis set, and a Davidson iteration count. When FCI is reexecuted for the same molecule and basis set, it automatically restarts from the latest Davidson iteration by identifying the saved files and loading the intermediate result from them.

3. Results and Discussion

In this section, we evaluate the execution time and scalability of our optimized distributed FCI implementation by comparing it with a naive combination of prior works. Then, we also validate its exactness by comparing the calculated energies between our implementation and PySCF. Finally, we discuss the time overhead of the restart function of our implementation. In addition, we calculate the ground-state energies of 136 combinations of molecules and basis sets with the molecular geometries obtained from Computational Chemistry Comparison and Benchmark DataBase (CCCBDB) .57 All of the results are listed in the Supporting Information.

For our evaluation, we use the “V-nodes” of the AI Bridging Cloud Infrastructure (ABCI) supercomputer.58 Each node contains two Intel Xeon Gold 6148 CPUs, 384 GB of memory, a 1.6 TB local NVMe SSD, and two InfiniBand EDR cards. In all of the evaluations with multiple processes, we run two processes on each node. Each process operates with 40 threads on each CPU, which has 40 logical cores with Hyper-Threading technology enabled. In addition, the 7.68 TB network-attached NVMe HDD is used to save data for restarts.

3.1. Time Evaluation

We compare the overall execution time of FCI between the naive combination of prior works17,39 and our optimized hybrid MPI-OpenMP implementation for BH3/cc-pVDZ, which has 5.6 × 108 determinants. 64 processes run on 32 servers in this experiment. As shown in Figure 7, our optimized implementation reduces the overall time from 331 to 150 s (by 55%). While the naive combination takes 95.6 s to reset and add the intermediate buffers (light blue), our cyclic thread-safe data assignment method completely eliminates them and takes only 1.6 s to reset buffers for MPI communication. Moreover, our MPI implementation with Allgather and Alltoall significantly reduces the interprocess communication time (dark blue plus orange) from 139 to 56 s compared to the naive combination of Allgatherv and Reduce and Scatter.

Figure 7.

Figure 7

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Overall execution time of FCI for BH3/cc-pVDZ with the naive combination of prior works17,39 and our optimized implementation. 64 processes run on 32 servers.

3.2. Scalability Evaluation

Next, we compare the strong scalability for LiH/cc-pVQZ with 1.3 × 107 determinants and NaBH4/STO-3G with 1.9 × 109 determinants. Figure 8 shows the speedup with multiple processes over a single process of our optimized implementation and the naive combination of prior works.17,39 We can see that our implementation achieves higher scalability than the naive combination. Especially, our implementation scales to 512 processes running on 256 nodes for LiH/cc-pVQZ where the amount of interprocess communication is relatively small. Moreover, it also scales to 128 processes running on 64 nodes for NaBH4/STO-3G, where the amount of interprocess communication is relatively large, whereas the naive combination fails to scale with more than 32 processes.

Figure 8.

Figure 8

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Strong scalability to the number of processes for LiH/cc-pVQZ and NaBH4/STO-3G. The y-axis indicates the speedup with n processes over a single process, and the red dotted line shows the ideal n times speedup.

3.3. Exactness Validation

To validate the exactness of our hybrid FCI implementation, we compare the ground-state energies of 105 combinations of molecules and basis sets with those of the PySCF implementation. The largest energy difference is only 8.27 × 10–10, which is negligible for FCI calculation. The detailed results are provided in the Supporting Information.

3.4. Restart Time Evaluation

The time taken to load the intermediate results from specific files for restart depends on the number of processes and the size of C and σ. The largest size evaluated in this work is 19 TB for C3H8/STO-3G with 512 processes, where the overall execution time of FCI is 113.6 h. For this experiment, we restart the FCI execution four times, and the total restart time is 3342 s. Thus, the time overhead of the restarts is only 0.8%. To validate the exactness of FCI across restarts, we compare the energies between the cases with four restarts and no restart for C2H4O2/STO-3G having 5.4 × 1011 determinants and confirm that the energy difference is only 2 × 10–11.

4. Conclusions

In this work, we present a hybrid MPI-OpenMP implementation of the FCI to expand its applicability to large molecules. We apply a cyclic thread-safe data assignment method to eliminate intermediate buffers for efficient multithreading and optimize interprocess communication using the Allgather and Alltoall functions in MPI. Compared to the naive combination of prior MPI implementations, our optimized hybrid implementation reduces the FCI execution time by 55% with 64 processes and achieves a higher scalability to the number of processes. Consequently, we successfully performed the FCI calculation for C3H8/STO-3G with 1.3 trillion determinants, which is the largest scale to our knowledge. Furthermore, we provide the exact ground-state energies of 136 combinations of molecules and basis sets calculated with our implementation. We hope it will be helpful for future development and evaluation of emerging approximate methods in QCC.

Acknowledgments

The authors thank the ABCI ground challenge committee for providing an experimental environment and Associate Professor Ishikawa Atsushi of the Tokyo Institute of Technology for valuable advice.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.3c01190.

  • Ground-state energies calculated with the PySCF implementation and our hybrid MPI-OpenMP implementation of FCI and geometries of calculated molecules (PDF)

The authors declare no competing financial interest.

Supplementary Material

ct3c01190_si_001.pdf (232.3KB, pdf)

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