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arXiv:2604.26314

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

Sorin Bolos

quant-phphysics.chem-phSubmitted Apr 29, 2026• Announced Apr 30, 2026• 0 citations• Revised Apr 30, 2026 (v2)
v1 · Apr 29, 2026v2 · Apr 30, 2026
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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 $p_d(x) e^{-ζx}$, 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 $\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.

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Preparing matrix product states (MPS) on a quantum device is a key subroutine in many quantum algorithms. The most competitive methods, based on the renormalisation group, prepare translationally invariant MPS of size $L$ and bond dimension $χ$, up to an error $\varepsilon$, in circuit depth $\tilde O(χ^{4}\log(L/\varepsilon))$ or $\tilde O(χ^{6}\log\log(L/\varepsilon))$. We improve multiple aspects of these methods. First, using block-encoded correction maps, whose post-selection succeeds with constant probability, we render the preparation exact without sacrificing the scaling in $L$. Second, through a generalisation of oblivious amplitude amplification to isometries, we reduce the bond-dimension dependence, improving the depth to $\tilde O(χ^{2}\log L + χ^{4})$ or $\tilde O(χ^{2}\log\log L + χ^{4})$, and even to $\tilde O(χ^{3}\log L)$ for incoherent preparations. Finally, we extend the framework to non-translationally invariant MPS and prove logarithmic-depth exact preparation for independent and identically distributed random tensor sequences. Confirmed by numerical studies, these results constitute, to the best of our knowledge, the most efficient exact MPS preparation protocols in the relevant parameter regimes.

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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 pd(x)e−ζxp_d(x) e^{-ζx}pd​(x)e−ζx, we derive analytical MPS constructions with constant bond dimension χ=d+1χ= d + 1χ=d+1, requiring O(n)O(n)O(n) classical and quantum resources for nnn-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 5qubits 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 18qubits. 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 10−610^{-6}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.
Quantum computers promise to solve the electronic structure problem for a large class of molecules. However, the performance of relevant quantum algorithms hinges on preparing initial states with substantial overlap with the target eigenvector. For classically challenging molecules with strong electron correlation, starting from multi-reference states, such as complete active space (CAS) wavefunctions is necessary. Unfortunately, the most advanced state preparation protocols applied to such states result in a gate complexity that scales exponentially with the active space size ddd. In fact, even encoding a CAS state classically is traditionally believed to be intractable for chemically relevant systems. Here, we draw insights from the recently introduced Quantum Paldus Transform (QPT) to show that there exists an efficient classical representation of CAS states and to design a new state preparation routine outperforming previous ones. The QPT represents a transformation from the Fock basis to a friendlier symmetry-adapted basis. Our main contribution consists in showing that CAS states expanded in this basis can efficiently be represented as a matrix product state (MPS) with a bond dimension scaling as O(d2)O(d^2)O(d2). One can then efficiently load the MPS on a quantum computer and use the inverse QPT to transform the state to the Fock basis. Moreover, our method can easily be extended to the efficient preparation of CAS states in first quantisation with similar complexity. Crucially, we demonstrate that the complexity of both state preparation protocols only grows polynomially as O(d3)O(d^3)O(d3) , which constitutes to the best of our knowledge an exponential improvement over the state of the art.
The amplitude encoding of an arbitrary nnn-qubit state vector requires Ω(2n)Ω(2^n)Ω(2n) gate operations, owing to the exponential dimension of the Hilbert space. We can, however, form dimensionality-reduced representations of quantum states using matrix product states (MPS). In this article, we illustrate that MPS techniques enable the preparation of quantum states representative of functions with complexity up to low-degree piecewise polynomials via shallow-depth quantum circuits with accuracy exceeding 99.99\%. We extend these results to the approximate amplitude encoding of pixel values. We showcase this approach by efficiently preparing a 128×128128\times 128128×128 ChestMNIST medical image (https://medmnist.com/) on 14 qubits with fidelity exceeding 99.2\% on a circuit with a total depth of just 425 single-qubit rotation and CNOT gates.
Quantum computation offers significant potential for accelerating the simulation of molecules and materials through algorithms such as quantum phase estimation (QPE). However, the expected speedup in ground-state energy estimation depends critically on the ability to efficiently prepare an initial state with high overlap with the true ground state. For strongly correlated molecules such as iron-sulfur clusters, this overlap is demonstrated to decay exponentially with system size. To alleviate this problem, we introduce an efficient classical algorithm to find entanglement-minimized orbitals (EMOs) and their orderings using spin-adapted low-bond-dimension matrix product states (MPS). The EMO basis yields a more compact ground-state representation, significantly easing initial state preparation for challenging systems. Our algorithm improves initial state overlap by nearly an order of magnitude over prior orbital optimization approaches for an iron-sulfur cluster with four irons, and is scalable to larger systems with many unpaired electrons, including the P-cluster and FeMo-cofactor in nitrogenase with eight transition metal centers. For these systems, we achieve substantial enhancements on initial state overlap by factors of O(10^{2}) and O(10^{5}), respectively, compared to results obtained using localized orbitals. Our results show that initial state preparation for these challenging systems requires far fewer resources than prior estimates suggested.
Loading functions into quantum computers represents an essential step in several quantum algorithms, such as quantum partial differential equation solvers. Therefore, the inefficiency of this process leads to a major bottleneck for the application of these algorithms. Here, we present and compare two efficient methods for the amplitude encoding of real polynomial functions on n qubits. This case holds special relevance, as any continuous function on a closed interval can be uniformly approximated with arbitrary precision by a polynomial function. The first approach relies on the matrix product state representation (MPS). We study and benchmark the approximations of the target state when the bond dimension is assumed to be small. The second algorithm combines two subroutines. Initially we encode the linear function into the quantum registers either via its MPS or with a shallow sequence of multi-controlled gates that loads the linear function's Hadamard-Walsh series, and we explore how truncating the Hadamard-Walsh series of the linear function affects the final fidelity. Applying the inverse discrete Hadamard-Walsh transform converts the state encoding the series coefficients into an amplitude encoding of the linear function. Thus, we use this construction as a building block to achieve an exact block encoding of the amplitudes corresponding to the linear function on k0 qubits and apply the quantum singular value transformation that implements a polynomial transformation to the block encoding of the amplitudes. This unitary together with the Amplitude Amplification algorithm will enable us to prepare the quantum state that encodes the polynomial function on k0 qubits. Finally we pad n−k0 qubits to generate an approximated encoding of the polynomial on n qubits, analyzing the error depending on k0 . In this regard, our methodology proposes a method to improve the state-of-the-art complexity by introducing controllable errors.
Gausslet orbitals on a cubic lattice, introduced in [Steven R. White, J. Chem. Phys. 147, 244102 (2017)], represent a localized, smooth, and systematically refineable basis set featuring a "diagonal" approximability of the electron repulsion integral tensor. The present work develops efficient algorithms for evaluating overlap integrals required for electronic-structure simulations using these Gausslets, specifically the kinetic, nuclear, and electron repulsion integrals (both with and without the diagonal approximation). Computational efficiency improvements rest on the exploitation of translation, permutation, and octahedral symmetries, a reordering and precomputation of nested sums, as well as an early truncation of small coefficients. Our algorithms reduce the number of electron repulsion integrals on a 5×5×55 \times 5 \times 55×5×5 grid from 1254=244140625125^4 = 2441406251254=244140625 to 138187138187138187 due to symmetries, and achieve a wall-clock runtime for evaluating the remaining integrals with a truncation tolerance of 10−510^{-5}10−5 in under 1 second on a laptop computer. We apply the developed methodology to compute the ground state of the hydrogen atom and molecule as a demonstration.
Preparing matrix product states (MPS) on a quantum device is a key subroutine in many quantum algorithms. The most competitive methods, based on the renormalisation group, prepare translationally invariant MPS of size LLL and bond dimension χχχ, up to an error ε\varepsilonε, in circuit depth O~(χ4log⁡(L/ε))\tilde O(χ^{4}\log(L/\varepsilon))O~(χ4log(L/ε)) or O~(χ6log⁡log⁡(L/ε))\tilde O(χ^{6}\log\log(L/\varepsilon))O~(χ6loglog(L/ε)). We improve multiple aspects of these methods. First, using block-encoded correction maps, whose post-selection succeeds with constant probability, we render the preparation exact without sacrificing the scaling in LLL. Second, through a generalisation of oblivious amplitude amplification to isometries, we reduce the bond-dimension dependence, improving the depth to O~(χ2log⁡L+χ4)\tilde O(χ^{2}\log L + χ^{4})O~(χ2logL+χ4) or O~(χ2log⁡log⁡L+χ4)\tilde O(χ^{2}\log\log L + χ^{4})O~(χ2loglogL+χ4), and even to O~(χ3log⁡L)\tilde O(χ^{3}\log L)O~(χ3logL) for incoherent preparations. Finally, we extend the framework to non-translationally invariant MPS and prove logarithmic-depth exact preparation for independent and identically distributed random tensor sequences. Confirmed by numerical studies, these results constitute, to the best of our knowledge, the most efficient exact MPS preparation protocols in the relevant parameter regimes.