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This paper introduces a method for amplitude encoding Slater-Type Orbitals (STOs) on quantum computers using Matrix Product States (MPS), enabling efficient representation of atomic wavefunctions. They derive analytical MPS constructions with constant bond dimension for 1D STOs, requiring only O(n) resources, and extend this to 3D, demonstrating bounded encoding complexity. The method is validated by computing one-electron integrals on IBM Heron hardware and multi-center overlap integrals, showcasing the feasibility of using STOs in quantum computational chemistry.
Forget exponentially scaling complexity: representing realistic atomic orbitals on quantum computers might be easier than we thought, thanks to bounded entanglement in Matrix Product State encodings.
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 $p_d(x) e^{-味x}$, we derive analytical MPS constructions with constant bond dimension $蠂= d + 1$, 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.
