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arXiv:2511.13677 · The Journal of Chemical Physics

Open-shell frozen natural orbital approach for quantum eigensolvers

Angela F. Harper, Xiaobing Liu, Scott N. Genin, Ilya G. Ryabinkin

Published Apr 15, 2026• 0 citations
arXivHTMLPDF

Abstract

We present an open-shell frozen natural orbital (FNO) approach, which utilizes the second-order Z-averaged perturbation theory (ZAPT2), to reduce the restricted open-shell Hartree-Fock virtual space size with controllable accuracy. Our ZAPT2 frozen natural orbital (ZAPT-FNO) selection scheme significantly outperforms the canonical molecular orbital virtual space truncation scheme based on Hartree-Fock orbital energies, especially when using large multiple-polarized and augmented basis sets. We demonstrate that the ZAPT-FNO-selected virtual orbitals lead to a systematic convergence of the correlation energies but, more importantly, to the singlet-triplet T1-S0 energy gaps with respect to the complete active space (CAS) [occupied + virtual] size. We confirm our findings by simulating T1-S0 gaps in H2O2 and O2 molecules using the traditional complete active space configuration interaction (CASCI) approach, as well as in stretched CH2, for which we also employed the iterative qubit coupled cluster (iQCC) method as a quantum eigensolver. Finally, we applied the iQCC method with a ZAPT-FNO-selected active space to the phosphorescent Ir(ppy)3 complex with 260 electrons, where extended basis sets are required to achieve chemical (∼1 mEh) accuracy. In this case, CASCI results are not available; however, the iQCC-computed T1-S0 gaps show robust convergence with an enlarging basis set and CAS size, approaching the experimental value. Thus, the ZAPT-FNO method is very promising for improving the accuracy of quantum chemical modeling in a resource-efficient manner and opens the door to simulating open-shell states of large materials within realistic active space sizes and without compromising on basis-set quality.

DOI
10.1063/5.0312011

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We present an open-shell frozen natural orbital (FNO) approach, which utilizes the second-order Z-averaged perturbation theory (ZAPT2), to reduce the restricted open-shell Hartree-Fock virtual space size with controllable accuracy. Our ZAPT2 frozen natural orbital (ZAPT-FNO) selection scheme significantly outperforms the canonical molecular orbital virtual space truncation scheme based on Hartree-Fock orbital energies, especially when using large multiple-polarized and augmented basis sets. We demonstrate that the ZAPT-FNO-selected virtual orbitals lead to a systematic convergence of the correlation energies but, more importantly, to the singlet-triplet T1-S0 energy gaps with respect to the complete active space (CAS) [occupied + virtual] size. We confirm our findings by simulating T1-S0 gaps in H2O2 and O2 molecules using the traditional complete active space configuration interaction (CASCI) approach, as well as in stretched CH2, for which we also employed the iterative qubit coupled cluster (iQCC) method as a quantum eigensolver. Finally, we applied the iQCC method with a ZAPT-FNO-selected active space to the phosphorescent Ir(ppy)3 complex with 260 electrons, where extended basis sets are required to achieve chemical (∼1 mEh) accuracy. In this case, CASCI results are not available; however, the iQCC-computed T1-S0 gaps show robust convergence with an enlarging basis set and CAS size, approaching the experimental value. Thus, the ZAPT-FNO method is very promising for improving the accuracy of quantum chemical modeling in a resource-efficient manner and opens the door to simulating open-shell states of large materials within realistic active space sizes and without compromising on basis-set quality.
Projection-based embedding offers a robust framework for modeling chemical processes in large molecular systems. It partitions computations into low-level treatments for the environmental subsystems and high-level treatments for the active subsystems. To address the computational bottleneck of applying high-accuracy wavefunction theory to the active regions, we introduce frozen natural orbitals (FNOs) approach. This technique efficiently compresses the virtual orbital space to generate a compact set of natural orbitals, which significantly accelerates the convergence of correlation energy recovery. We evaluated the FNO-based embedding approach in comparison with two alternative virtual space truncation approaches for a diverse range of molecular systems. Our results demonstrate that the FNO-embedded method exhibits superior performance by delivering accurate correlation energies with a substantially fewer number of virtual orbitals. Its potential for quantum computing is also discussed, as this reduction in orbitals can be directly translated to lower qubit requirements and thereby facilitate the quantum simulations on near-term quantum hardware.
We present a unified frozen natural orbital (FNO) framework for periodic Gaussian-orbital post-Hartree-Fock calculations with explicit k-point sampling. The approach includes conventional ground-state FNOs together with state-specific (SS-FNO) and state-averaged (SA-FNO) variants constructed from perturbative one-particle density matrices. We implement these approximations for periodic Moller-Plesset perturbation theory, algebraic diagrammatic construction, and equation-of-motion coupled cluster methods in the PySCF software package and benchmark them for correlation energies, equations of state, fundamental band gaps, and quasiparticle band structures of semiconductors and insulators. FNO truncation reproduces canonical results with small errors while substantially reducing computational cost. SS-FNO enables accurate large-basis band-gap calculations, including quadruple-zeta results, whereas SA-FNO efficiently compresses the virtual space for multi-state band-structure calculations. Combined with basis-set and thermodynamic-limit extrapolations, the framework provides a practical route to high-accuracy correlated calculations in periodic systems.
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.
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.
Nuclear quantum effects such as zero-point energy and hydrogen tunneling play a central role in many biological and chemical processes. The nuclear-electronic orbital (NEO) approach captures these effects by treating selected nuclei quantum mechanically on the same footing as electrons. On classical computers, the resources required for an exact solution of NEO-based models grow exponentially with system size. By contrast, quantum computers offer a means of solving this problem with polynomial scaling. However, due to the limitations of current quantum devices, NEO simulations are confined to the smallest systems described by minimal basis sets, whereas realistic simulations beyond the Born–Oppenheimer approximation require more sophisticated basis sets. For this purpose, we herein extend a hardware-efficient ADAPT-VQE method to the NEO framework in the frozen natural orbital (FNO) basis. We demonstrate on H 2 and D 2 molecules that the NEO-FNO-ADAPT-VQE method reduces the CNOT count by several orders of magnitude relative to the NEO unitary coupled cluster method with singles and doubles while maintaining the desired accuracy. This extreme reduction in the CNOT gate count is sufficient to permit practical computations employing the NEO method─an important step toward accurate simulations involving nonclassical nuclei and non-Born–Oppenheimer effects on near-term quantum devices. We further show that the method can capture isotope effects, and we demonstrate that inclusion of correlation energy systematically improves the prediction of difference in the zero-point energy (ΔZPE) between isotopes.
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.