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This paper introduces a biquaternion algebra framework that enhances relativistic electronic structure calculations by utilizing bilinear multiplication to optimize matrix operations. The authors developed HMATLIB, a matrix library that significantly reduces the computational complexity of matrix-valued biquaternion multiplication, cutting the number of required real matrix multiplications from 64 to 24. Benchmark results show that this approach accelerates computations on CPU and hybrid CPU/GPU systems, achieving speedups of up to 61 times over traditional methods, thereby establishing biquaternion algebra as a superior alternative for electronic structure calculations.
Biquaternion algebra slashes computational costs in relativistic electronic structure calculations, achieving up to 61x speedups over conventional methods.
Quaternion algebra provides a natural representation of time-reversal-symmetric matrix structures in relativistic electronic-structure theory, whereas complementary time-reversal-antisymmetric structures extend this representation to complex quaternions, or biquaternions. Here, we explicitly exploit biquaternion algebra for fundamental objects, including operators, kinetically and magnetically balanced basis functions, and their expectation values, within a unified framework encompassing real, complex, and real quaternion subalgebras as special cases. To realize this framework computationally, we have developed HMATLIB, a biquaternion matrix library implemented within the ReSpect package for pure CPU and hybrid CPU/GPU execution. A key development is a bilinear algorithm for matrix-valued biquaternion multiplication that reduces the number of real matrix--matrix multiplications from 64 to 24 compared with the conventional component-wise approach, while the biquaternion representation effectively doubles the maximum accessible matrix dimension under the same 32-bit indexing constraint. Numerical benchmarks on modern CPU and GPU architectures demonstrate that the biquaternion formulation consistently outperforms its isomorphic complex-algebra counterpart. For the largest matrices reported, hybrid CPU/GPU execution accelerates bilinear matrix multiplication by approximately 5-8 times over pure CPU execution and matrix diagonalization by approximately 12 and 61 times relative to CPU oneAPI MKL and NVHPC OpenBLAS, respectively. These results establish biquaternion algebra as a general and computationally advantageous framework for modern relativistic electronic-structure calculations.