This paper presents a novel quantum computing framework, termed "Adaptive Tensor Aggregation" (ATA), designed to accelerate quantum machine learning algorithms by leveraging adaptive tensor aggregation. Traditional quantum computation often struggles with the limitations of limited qubit resources, necessitating strategies for efficient tensor manipulation. ATA addresses these challenges through a dynamic adjustment of qubit count and state representation, guided by quantum entanglement and superposition. The core mechanism centers around a novel algorithm that iteratively adjusts the tensor aggregation level based on the current state of the system, optimizing for computational complexity and resource utilization. This approach allows for the efficient execution of complex tensor operations, crucial for many machine learning tasks. The proposed framework demonstrates significant improvements in performance compared to existing quantum algorithms, particularly in the context of large-scale tensor computations.
Electron-repulsion integrals describe the Coulomb interaction between charge distributions built from orbital basis functions. Most integral algorithms generate these quantities through Cartesian Gaussian functions, whose angular shapes are written as powers of
x,
y, and
z, and then transform the result to spherical functions. This route is effective, but from
d shells onward the Cartesian representation contains more functions than the spherical space required by the calculation. Direct real or complex solid-harmonic engines work in that target space from the beginning. They therefore produce a smaller final Coulomb tensor while preserving the ordering, phase, and magnetic-quantum-number labels that describe its angular structure. Following this structure beyond integral evaluation reveals direct connections to the algorithms that use the tensor. Simple analytical counts quantify tensor size, angular blocks, radial Slater--Condon parameters, and pair-space work. These quantities guide low-rank factorization, local Hamiltonian construction, quantum simulation, and transformations to spinor or effective-model bases. In this way, solid-harmonic integral engines provide a direct bridge between efficient integral generation and structured many-electron computation.