Amplitude Encoding of Slater-Type Orbitals via Matrix Product States: Efficient State Preparation and Integral Evaluation on Quantum Hardware

Slater-type orbitals (STOs) provide the physically correct description of atomic wavefunctions but have been largely replaced by Gaussian-type orbitals in computational chemistry due to the lack of closed-form multi-center integrals. We present a systematic study of amplitude encoding of STOs on quantum computers using matrix product states (MPS). For one-dimensional orbital functions of the form p_d(x) e^-ζx, we derive analytical MPS constructions with constant bond dimension χ= d + 1, requiring O(n) classical and quantum resources for n-qubit registers with no grid sampling. We demonstrate a complete one-electron integral pipeline - overlap, kinetic energy, and nuclear attraction - in one dimension, validating the overlap and kinetic energy on IBM Heron processors at 5 qubits with 0.67% hardware-induced error using Zero-Noise Extrapolation. In three dimensions, we compute multi-center overlap integrals between 1s and 2s orbitals in Cartesian coordinates with 0.02% discretization error at 18 qubits. A systematic entanglement analysis reveals that the MPS bond dimension of three-dimensional STOs in Cartesian coordinates saturates with increasing grid resolution - reaching sim138 for the hydrogen 1s orbital at 12 qubits per coordinate - establishing bounded encoding complexity rather than the exponential scaling initially expected. The SVD truncation threshold provides a practical resource parameter, reducing the bond dimension to 39 at threshold 10^-6 with negligible accuracy loss. These results map the entanglement landscape for amplitude encoding of atomic orbitals and establish MPS-based state preparation as a viable path toward exact STO basis sets on quantum computers.