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arXiv:2509.05733 · Journal of Chemical Theory and Computation

Improving the Runtime of Quantum Phase Estimation for Chemistry through Basis Set Optimization

Pauline J. Ollitrault, Jérôme F. Gonthier, Dario Rocca, Gian-Luca Anselmetti, Matthias Degroote, Nikolaj Moll, Raffaele Santagati, Michael Streif

Published Nov 20, 2025• 2 citations
arXivHTMLPDF

Abstract

High Resolution Image Download MS PowerPoint Slide Quantum phase estimation (QPE) is a promising quantum algorithm for obtaining molecular ground-state energies with chemical accuracy. However, its computational cost, dominated by the Hamiltonian 1-norm λ and the cost of the block encoding, scales at least quadratically with the number of molecular orbitals, making it challenging to incorporate dynamic correlation by enlarging the active space. In this work, we investigate two strategies to mitigate this cost through the optimization of the basis set. First, we investigate whether adjusting the coefficients of Gaussian basis functions can minimize the 1-norm while preserving the accuracy of the ground state energy. Although this method leads to a reduction in the 1-norm up to 10%, this reduction is system-dependent and diminishes with increasing molecular size. Second, we demonstrate that employing a large-basis-set frozen natural orbital (FNO) strategy results in a substantial reduction in QPE resources without compromising accuracy. We study a data set of 58 small organic molecules and the dissociation curve of N 2, and demonstrate that an active space constructed from orbitals derived from larger basis sets captures correlation effects more effectively. This approach yields up to an 80% reduction in the 1-norm λ and also leads to a 55% reduction in the number of orbitals. Our results highlight that improving the quality, not just the size, of the orbital basis is a viable strategy for extending QPE to include dynamical correlation, making progress toward scalable and chemically accurate quantum simulations with tractable resource requirements.

DOI
10.1021/acs.jctc.5c01512

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High Resolution Image Download MS PowerPoint Slide Quantum phase estimation (QPE) is a promising quantum algorithm for obtaining molecular ground-state energies with chemical accuracy. However, its computational cost, dominated by the Hamiltonian 1-norm λ and the cost of the block encoding, scales at least quadratically with the number of molecular orbitals, making it challenging to incorporate dynamic correlation by enlarging the active space. In this work, we investigate two strategies to mitigate this cost through the optimization of the basis set. First, we investigate whether adjusting the coefficients of Gaussian basis functions can minimize the 1-norm while preserving the accuracy of the ground state energy. Although this method leads to a reduction in the 1-norm up to 10%, this reduction is system-dependent and diminishes with increasing molecular size. Second, we demonstrate that employing a large-basis-set frozen natural orbital (FNO) strategy results in a substantial reduction in QPE resources without compromising accuracy. We study a data set of 58 small organic molecules and the dissociation curve of N 2, and demonstrate that an active space constructed from orbitals derived from larger basis sets captures correlation effects more effectively. This approach yields up to an 80% reduction in the 1-norm λ and also leads to a 55% reduction in the number of orbitals. Our results highlight that improving the quality, not just the size, of the orbital basis is a viable strategy for extending QPE to include dynamical correlation, making progress toward scalable and chemically accurate quantum simulations with tractable resource requirements.
The variational quantum eigensolver (VQE) has emerged as one of the leading quantum algorithms for solving electronic structure problems on near-term noisy intermediate-scale quantum devices. However, its practical application to quantum chemistry remains challenging due to the limited coherence time, imperfect quantum gate fidelity, and the large number of measurements required, which together confine current electronic structure simulations to relatively small active spaces. In this work, we present an orbital-optimized VQE framework based on orbital compression, designed to improve the accuracy of electronic structure calculations while maintaining relatively small active spaces. Frozen natural orbitals (FNO) and split virtual orbitals (SVO) are first employed to construct compact active spaces for VQE simulations, leading to the FNO/SVO-VQE approach. Orbital optimization is then incorporated to further recover electron correlation effects, resulting in the FNO/SVO-OO-VQE methods. We apply the proposed method to simulate potential energy surfaces for molecular dissociation and the activation energy of formaldehyde decomposition. Numerical results demonstrate that both FNO-OO-VQE and SVO-OO-VQE improve the variational accuracy while substantially reducing measurement cost.
In spite of recent advances, quantum computers are expected to be sufficiently noisy in the coming few years to the extent of limiting quantum chemical calculations to relatively small number of orbitals. However, even with reasonable quality single particle basis sets, small active spaces with limited orbitals can result in a significant fraction of correlation energy being lost, motivating the design of moderate quality qubit-efficient basis sets for quantum algorithms. We begin by reoptimizing the existing minimal basis sets using a genetic algorithm-inspired approach in conjunction with aggressive refinement strategies, and generate modified minimal basis sets (MSTO-kG basis; k = 2-11) for atoms from H through F. The ground state energies of H through F using our MSTO bases at FCI level of theory yield ground state energies that are comparable or sometimes even lower than those obtained using 6-31G basis sets. In the case of Li, the MSTO bases surpass the performance of cc-pVQZ bases. Thus, we obtain better atomic energies with same number of qubits relative to STO bases, and better/comparable energies with fewer qubits relative to higher quality bases. In the case of molecules, H2 performs poorly; a finding that is consistent with an earlier work in literature. For other molecules, Li2, C2, LiH, BeH and BeH2, the FCI results (except C2 for which we employ CISD) from our bases are comparable to/outperform those from 6-31G basis. Finally, we compare the resources required between different bases and find that MSTO bases yield better energies than the competing basis sets while incurring fewer qubits and two-qubit gates with VQE, QPE, and HHL. The logical T-gate counts are also found to be considerably lower for QPE and HHL respectively. Overall, our work paves way for more accurate yet less qubit-hungry quantum chemical calculations using near-term quantum computers.
Quantum simulation of molecular electronic structure is one of the most promising applications of quantum computing. However, achieving chemically accurate predictions for strongly correlated systems requires quantum phase estimation (QPE) on fault-tolerant quantum computing (FTQC) devices. Existing resource estimates for typical FTQC architectures suggest that such calculations demand millions of physical qubits, thereby placing them beyond the reach of near-term devices. Here, we investigate the feasibility of performing QPE for chemically relevant molecular systems in an early-FTQC regime, characterized by partial fault tolerance, constrained qubit budgets, and limited circuit depth. Our framework is based on single-ancilla, Trotter-based QPE implementations combined with partially randomized time evolution. Within this framework, we develop a novel Hamiltonian optimization strategy, termed unitary weight concentration, that reduces algorithmic cost by reshaping linear-combination-of-unitaries representations. Applying this framework to active-space models of iron-sulfur clusters, cytochrome P450 active sites, and CO2_22​-utilization catalysts, we perform end-to-end resource estimation using the latest version of the space-time efficient analog rotation (STAR) architecture. Our results indicate that ground-state energy estimation for active spaces of approximately 20 to 50 spatial orbitals, well beyond the reach of classical full configuration interaction, is achievable using ∼105\sim 10^5∼105 physical qubits, with runtimes on the order of days to weeks. These findings demonstrate that while full-fledged fault-tolerant quantum computers will be required for even larger molecular simulations, chemically meaningful quantum chemistry problems are already within reach in an experimentally relevant, early-FTQC regime.
The computational cost of quantum algorithms for physics and chemistry is closely linked to the spectrum of the Hamiltonian, a property that manifests in the necessary rescaling of its eigenvalues. The typical approach of using the 1-norm as an upper bound to the spectral norm to rescale the Hamiltonian suits the most general case of bounded Hermitian operators but neglects the influence of symmetries commonly found in chemical systems. In this work, we introduce a hierarchy of symmetry-aware spectral bounds that provide a unified understanding of the performance of quantum phase estimation algorithms using block-encoded electronic structure Hamiltonians. We present a variational and numerically tractable method for computing these bounds, based on orbital optimization, to demonstrate that the computed bounds are smaller than conventional spectral bounds for a variety of molecular benchmark systems. We also highlight the unique analytical and numerical scaling behavior of these bounds in the thermodynamic and complete basis set limits. Our work shows that there is room for improvement in reducing the 1-norm, not yet achieved through methods like double factorization and tensor hypercontraction, but highlights the potential challenges in improving the performance of current quantum algorithms beyond small constant factors through 1-norm reduction techniques alone.
The method of increments and frozen natural orbital (MI-FNO) framework is introduced to help expedite the application of noisy, intermediate-scale quantum (NISQ) devices for quantum chemistry simulations. The MI-FNO framework provides a systematic reduction of the occupied and virtual orbital spaces for quantum chemistry simulations. The correlation energies of the resulting increments from the MI-FNO reduction can then be solved by various algorithms, including quantum algorithms such as the phase estimation algorithm and the variational quantum eigensolver (VQE). The unitary coupled-cluster singles and doubles VQE framework is used to obtain correlation energies for the case of small molecules (i.e., BeH2, CH4, NH3, H2O, and HF) using the cc-pVDZ basis set. The quantum resource requirements are estimated for a constrained geometry complex catalyst that is utilized in industrial settings for the polymerization of α-olefins. We show that the MI-FNO approach provides a significant reduction in the quantum bit (qubit) requirements relative to the full system simulations. We propose that the MI-FNO framework can create scalable examples of quantum chemistry problems that are appropriate for assessing the progress of NISQ devices.
Quantum simulations of molecular systems on quantum computers often employ minimal basis sets of Gaussian orbitals. In comparison with more realistic basis sets, quantum simulations employing minimal basis sets require fewer qubits and quantum gates but yield results of lower accuracy. A natural strategy to achieve more accurate results is to increase the basis set size, which, in turn, requires increasing the number of qubits and quantum gates. Here we explore the use of intrinsic atomic orbitals in quantum simulations of molecules to improve the accuracy of energies and properties at the same computational cost required by a minimal basis. We investigate ground-state energies and one- and two-body density operators in the framework of the variational quantum eigensolver employing and comparing different Ans\"atze. We also demonstrate the use of this approach in the calculation of ground- and excited-state energies of small molecules by a combination of quantum algorithms using IBM quantum computers.