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

TL;DR

This paper develops efficient matrix product state methods for amplitude encoding of Slater-type orbitals on quantum computers, enabling accurate integral evaluation and state preparation with bounded entanglement complexity.

Contribution

It introduces analytical MPS constructions for STOs, demonstrates quantum hardware validation, and analyzes entanglement to establish scalable encoding strategies.

Findings

Analytical MPS with constant bond dimension for 1D STOs

Validated overlap and kinetic energy calculations on IBM quantum hardware

Bounded entanglement complexity in 3D STO encoding with practical resource parameters

Abstract

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 $p_d(x) e^{-\zeta x}$, we derive analytical MPS constructions with constant bond dimension $\chi = d + 1$, 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 $\sim$138 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.