# GPU-Accelerated Fock Matrix Computation with Efficient Reduction
## 3 Related Works  
As described in Section 2, constructing the Fock matrix is a computationally demanding process involving complex molecular integrals and intricate summations. In the Hartree-Fock theory, iterative updates of the Fock matrix constitute the dominant computational bottleneck. This section briefly summarizes accumulated insights and implementation strategies for acceleration, focusing primarily on parallelization using CPUs and GPUs. The computational bottleneck in the calculation of the Fock matrix can be primarily attributed to two factors: the evaluation of ERIs (Section 2.2.1) and the subsequent aggregation of these values (Section 2.2.2). We particularly highlight the latter, directing attention to the efficient reduction of ERIs, while introducing previous research in this area. The 2009 study by [PERSON] and [PERSON] [12] is known as the first fully GPU-accelerated implementation of the Fock matrix construction within the Direct-SCF framework. Recognizing the significantly different memory access patterns, distinct algorithms were employed for the Coulomb \(\mathbf{J}\)- and exchange \(\mathbf{K}\)-matrices. In both algorithms, the ERIs are pre-sorted according to the upper bound derived from the Schwarz inequality, thereby enhancing the effectiveness of screening on the GPU. This approach does not fully exploit the eight-fold symmetry of ERIs in Equation (18). Consequently, computing the \(\mathbf{J}\)- and \(\mathbf{K}\)-matrices separately involves approximately two-times and four-times redundant integral evaluations, respectively. In 2012, [PERSON] and [PERSON] [13] proposed a hybrid approach using CPUs and GPUs for the Hartree-Fock method, with a focus on efficient Fock matrix construction. They employed a blocked Fock and density matrix representation to improve cache locality on the CPU and promote coalesced memory access on the GPU. While their approach fully exploits the eight-fold symmetry of ERIs, updates to the Fock matrix using atomic operations remain susceptible to memory contention issues. In 2017, [PERSON] et al. [15] proposed two techniques for aggregating the values of ERIs in the Fock matrix calculation. This paper discusses the acceleration of the Hartree-Fock method in the quantum chemistry package GAMESS [37], describing a CPU implementation using hybrid parallelization with OpenMP and MPI. The \"Private Fock\" algorithm allocates a unique Fock matrix for each CPU thread, reducing memory contention by multiple threads in Equation (24). Conversely, to avoid the increased memory footprint associated with per-thread Fock matrices, the \"Shared Fock\" algorithm was also employed, sharing a single Fock matrix. Both algorithms reduce the number of overhead-intensive atomic operations by performing partial ERI reductions thread-locally. However, in highly parallel GPU architectures, only a limited number of integrals can be locally aggregated, thereby involving frequent atomic additions across numerous GPU threads. In 2020, [PERSON] et al. [18] proposed methods for constructing the Fock matrix on shared-memory and distributed-memory parallel computers. Similar discussions to those by [PERSON] et al. were conducted, comparing two techniques: \"Block buffer accumulation\" and \"Strip buffer accumulation\", both involving thread-local partial copies of the \(\mathbf{J}\)- and \(\mathbf{K}\)-matrices. The former utilizes copies of only the six elements of Equation (24), while the latter employs larger partial copies to further reduce atomic operations. They reported a 36% reduction in computation time with the Strip buffer accumulation compared to the Block buffer accumulation. In the same year, [PERSON] et al. [19] proposed an accelerated Fock matrix construction algorithm for the Hartree-Fock method on GPUs. Their work detailed several optimization strategies. For ERI evaluation, integrals were classified by the type of angular momentum and contraction degree \(K\) in Equation (12), enabling dynamic load balancing through batched processing.