Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2311.02533 · Physical Review Applied

Ground-state-energy calculation for the water molecule on a superconducting quantum processor

Michael A. Jones, Harish J. Vallury, Lloyd C. L. Hollenberg

Published Jun 7, 2024• 8 citations
arXivHTMLPDF

Abstract

The accurate computation of properties of large molecular systems is classically infeasible and is one of the applications in which it is hoped that quantum computers will demonstrate an advantage over classical devices. However, due to the limitations of present-day quantum hardware, variational-hybrid algorithms introduced to tackle these problems struggle to meet the accuracy and precision requirements of chemical applications. Here, we apply the quantum computed moments (QCM) approach combined with a variety of noise-mitigation techniques to an eight qubit and spin-orbital representation of the water molecule (${\text{H}}_{2}\text{O}$). A noise-stable improvement on the variational result for a four-excitation trial state (circuit depth 25, 22 cz gates) was obtained, with the ground-state energy computed to be within $1.0\ifmmode\pm\else\textpm\fi{}0.8$ mHa of exact diagonalization in the 14 spin-orbital basis. Thus, the QCM approach, despite an increased number of measurements and noisy quantum hardware (cz error rates c.0.3% corresponding to expected error rates on the trial-state circuit of order 7%), is able to determine the ground-state energy of a nontrivial molecular system at the required accuracy (c.0.05%). To the best of our knowledge, these results are the largest calculations performed on a physical quantum computer to date to achieve millihartree agreement with exact diagonalization, and a promising indicator of how such hybrid approaches might scale to problems of interest in the low-error and fault-tolerant regimes as quantum computers develop.

DOI
10.1103/physrevapplied.21.064017

Related papers

Chemistry beyond the Hartree–Fock energy via quantum computed moments

Michael A. Jones, Harish J. Vallury, Charles D. Hill +1 more

Quantum computers hold promise to circumvent the limitations of conventional computing for difficult molecular problems. However, the accumulation of quantum logic errors on real devices represents a major challenge, particularly in the pursuit of chemical accuracy requiring the inclusion of electronic correlation effects. In this work we implement the quantum computed moments (QCM) approach for hydrogen chain molecular systems up to H[Formula: see text]. On a superconducting quantum processor, Hamiltonian moments, [Formula: see text] are computed with respect to the Hartree-Fock state, which are then employed in Lanczos expansion theory to determine an estimate for the ground-state energy which incorporates electronic correlations and manifestly improves on the direct energy measurement. Post-processing purification of the raw QCM data takes the estimate below the Hartree-Fock energy to within 99.9% of the exact electronic ground-state energy for the largest system studied, H[Formula: see text]. Calculated dissociation curves indicate precision at about 10mH for this system and as low as 0.1mH for molecular hydrogen, H[Formula: see text], over a range of bond lengths. In the context of stringent precision requirements for chemical problems, these results provide strong evidence for the error suppression capability of the QCM method, particularly when coupled with post-processing error mitigation. While calculations based on the Hartree-Fock state are tractable to classical computation, these results represent a first step towards implementing the QCM method in a quantum chemical trial circuit. Greater emphasis on more efficient representations of the Hamiltonian and classical preprocessing steps may enable the solution of larger systems on near-term quantum processors.

Scientific Reports•May 28, 2022•22 citations

Chemistry beyond the Hartree-Fock limit via quantum computed moments

Michael A. Jones, Harish J. Vallury, Charles D. Hill +1 more

Quantum computers hold promise to circumvent the limitations of conventional computing for difficult molecular problems. However, the accumulation of quantum logic errors on real devices represents a major challenge, particularly in the pursuit of chemical accuracy requiring the inclusion of dynamical effects. In this work we implement the quantum computed moments (QCM) approach for hydrogen chain molecular systems up to H$_6$. On a superconducting quantum processor, Hamiltonian moments, $\langle \mathcal{H}^p\rangle$ are computed with respect to the Hartree-Fock state, which are then employed in Lanczos expansion theory to determine an estimate for the ground-state energy which incorporates electronic correlations and manifestly improves on the variational result. Post-processing purification of the raw QCM data takes the estimate through the Hartree-Fock variational limit to within 99.9% of the exact electronic ground-state energy for the largest system studied, H$_6$. Calculated dissociation curves indicate precision at about 10mH for this system and as low as 0.1mH for molecular hydrogen, H$_2$, over a range of bond lengths. In the context of stringent precision requirements for chemical problems, these results provide strong evidence for the error suppression capability of the QCM method, particularly when coupled with post-processing error mitigation. Greater emphasis on more efficient representations of the Hamiltonian and classical preprocessing steps may enable the solution of larger systems on near-term quantum processors.

arXiv:2111.08132•Nov 15, 2021•0 citations

Implementing QESEM's High-Accuracy Error Mitigation on a Quantum Computer: a Water Potential Energy Surface Study

Renato Olarte Hernandez, Emanuele Rossi, Sonia Coriani +5 more

Quantum error mitigation (QEM) is essential for extracting chemically accurate results from near-term quantum hardware. Many widely used QEM methods rely on uncontrolled heuristics whose bias depends on the specific circuit and noise realization. In this work, we employ QESEM---a characterization-based, unbiased quasi-probabilistic mitigation method---on IBM's Aachen quantum processor to compute the ground-state potential energy surface (PES) of the symmetrically-stretched water molecule. We consider a classically-optimized, single-layer perfect-pairing tiled unitary product state ansatz. We map this ansatz to an 8-qubit register corresponding to a (4,4) active space and the STO-3G basis set. Compared to the statevector reference, we find that raw QPU results typically overestimate the ground-state energy by around 500~mHa across the scanned geometries. On the other hand, QESEM-mitigated results fall approximately within 100~mHa, 30~mHa, or ``chemical accuracy'' ($\sim$1.5~mHa), depending on the target precision. We benchmark QESEM at both loose (0.1~Ha) and tight (0.01~Ha) precision targets, evaluating both individual and merged batches of runs. As the precision target is tightened, the accuracy improves systematically, matching or exceeding results reported in the literature. We further quantify the sampling cost of these results, reporting the number of shots required at each precision level. Our results show how, with the current levels of hardware error, reaching the highest accuracies demands substantial QPU time. The results demonstrate that the characterization-based, unbiased error mitigation provided by QESEM allows to measure quantitatively meaningful potential energy surfaces on current quantum hardware. Concurrently, they highlight the sampling overhead associated with QEM, which remains a central bottleneck en route to larger chemical problems and higher precision.

quant-pharXiv:2609.07284•Sep 8, 2026•0 citations

Moments-based improved quantum computation of the electric dipole moment of molecular systems

Michael A. Jones, Harish J. Vallury, Manolo C. Per +2 more

With rapid progress being made in the development of platforms for quantum computation, there has been considerable interest in whether present-day and near-term devices can be used to solve problems of relevance. A commonly cited application area is the domain of quantum chemistry. While most experimental demonstrations of quantum chemical calculations on quantum devices have focused on the ground-state electronic energy of the system, other properties of the ground state, such as the electric dipole moment, are also of interest. Here, we employ the quantum-computed-moments method, based on the Lanczos cluster expansion, to estimate the dipole moment of the water molecule on an IBM Quantum superconducting quantum device. The noise-mitigated results agree with full-configuration-interaction calculations to within 0.03 ± 0.007 D ( 2 % ± 0.5 % ), compared with direct expectation-value determination (i.e., variational quantum eigensolver, VQE) with errors of the order of 0.07 D (5%), even when the VQE calculation is performed without noise. This demonstrates that moments-based energy-estimation techniques can be adapted to noise-robust evaluation of nonenergetic ground-state properties of chemical systems. We emphasize that the dipole moment is chosen as an example of such a property, but the method applies to any property determined by one- or two-body interactions.

arXiv:2509.10758•Physical Review Applied•Mar 19, 2026•0 citations

Simulating large-size quantum spin chains on cloud-based superconducting quantum computers

Hongye Yu, Yusheng Zhao, Tzu-Chieh Wei

Quantum computers have the potential to efficiently simulate large-scale quantum systems for which classical approaches are bound to fail. Even though several existing quantum devices now feature total qubit numbers of more than 100, their applicability remains plagued by the presence of noise and errors. Thus, the degree to which large quantum systems can successfully be simulated on these devices remains unclear. Here, we report on numerical results of physics-motivated variational ansatzes, as well as cloud simulations performed on several of IBM's superconducting quantum computers to simulate ground states of spin chains having a wide range of system sizes up to 102 qubits. Our numerical analysis shows that the accuracy of the ground-state energy and fidelity improves substantially by increasing the number of layers used in the ansatzes. From the cloud experiments, we find that the ground-state energies extracted from realizations across different quantum computers and system sizes reach the expected values to within errors that are small (i.e., on the percent level), including the inference of the energy density in the thermodynamic limit from these values. We achieve this accuracy through a combination of physics-motivated variational ansatzes, and efficient, scalable energy-measurement and error-mitigation protocols, including the use of a reference state in the zero-noise extrapolation. By using a 102-qubit system, we have been able to successfully apply up to 3186 CNOT gates in a single circuit when performing gate-error mitigation. Our accurate, error-mitigated results for random parameters in the ansatz states suggest that a standalone hybrid quantum-classical variational approach for large-scale $XXZ$ models considered in this work is feasible.

arXiv:2207.09994•Physical Review Research•Mar 16, 2023•36 citations

Noise-robust ground state energy estimates from deep quantum circuits

Harish J. Vallury, Michael A. Jones, Gregory A. L. White +3 more

In the lead up to fault tolerance, the utility of quantum computing will be determined by how adequately the effects of noise can be circumvented in quantum algorithms. Hybrid quantum-classical algorithms such as the variational quantum eigensolver (VQE) have been designed for the short-term regime. However, as problems scale, VQE results are generally scrambled by noise on present-day hardware. While error mitigation techniques alleviate these issues to some extent, there is a pressing need to develop algorithmic approaches with higher robustness to noise. Here, we explore the robustness properties of the recently introduced quantum computed moments (QCM) approach to ground state energy problems, and show through an analytic example how the underlying energy estimate explicitly filters out incoherent noise. Motivated by this observation, we implement QCM for a model of quantum magnetism on IBM Quantum hardware to examine the noise-filtering effect with increasing circuit depth. We find that QCM maintains a remarkably high degree of error robustness where VQE completely fails. On instances of the quantum magnetism model up to 20 qubits for ultra-deep trial state circuits of up to 500 CNOTs, QCM is still able to extract reasonable energy estimates. The observation is bolstered by an extensive set of experimental results. To match these results, VQE would need hardware improvement by some 2 orders of magnitude on error rates.

arXiv:2211.08780•Quantum•Sep 11, 2023•8 citations
The accurate computation of properties of large molecular systems is classically infeasible and is one of the applications in which it is hoped that quantum computers will demonstrate an advantage over classical devices. However, due to the limitations of present-day quantum hardware, variational-hybrid algorithms introduced to tackle these problems struggle to meet the accuracy and precision requirements of chemical applications. Here, we apply the quantum computed moments (QCM) approach combined with a variety of noise-mitigation techniques to an eight qubit and spin-orbital representation of the water molecule (H2O{\text{H}}_{2}\text{O}H2​O). A noise-stable improvement on the variational result for a four-excitation trial state (circuit depth 25, 22 cz gates) was obtained, with the ground-state energy computed to be within 1.0\ifmmode±\else\textpm\fi0.81.0\ifmmode\pm\else\textpm\fi{}0.81.0\ifmmode±\else\textpm\fi0.8 mHa of exact diagonalization in the 14 spin-orbital basis. Thus, the QCM approach, despite an increased number of measurements and noisy quantum hardware (cz error rates c.0.3% corresponding to expected error rates on the trial-state circuit of order 7%), is able to determine the ground-state energy of a nontrivial molecular system at the required accuracy (c.0.05%). To the best of our knowledge, these results are the largest calculations performed on a physical quantum computer to date to achieve millihartree agreement with exact diagonalization, and a promising indicator of how such hybrid approaches might scale to problems of interest in the low-error and fault-tolerant regimes as quantum computers develop.
Quantum computers hold promise to circumvent the limitations of conventional computing for difficult molecular problems. However, the accumulation of quantum logic errors on real devices represents a major challenge, particularly in the pursuit of chemical accuracy requiring the inclusion of electronic correlation effects. In this work we implement the quantum computed moments (QCM) approach for hydrogen chain molecular systems up to H[Formula: see text]. On a superconducting quantum processor, Hamiltonian moments, [Formula: see text] are computed with respect to the Hartree-Fock state, which are then employed in Lanczos expansion theory to determine an estimate for the ground-state energy which incorporates electronic correlations and manifestly improves on the direct energy measurement. Post-processing purification of the raw QCM data takes the estimate below the Hartree-Fock energy to within 99.9% of the exact electronic ground-state energy for the largest system studied, H[Formula: see text]. Calculated dissociation curves indicate precision at about 10mH for this system and as low as 0.1mH for molecular hydrogen, H[Formula: see text], over a range of bond lengths. In the context of stringent precision requirements for chemical problems, these results provide strong evidence for the error suppression capability of the QCM method, particularly when coupled with post-processing error mitigation. While calculations based on the Hartree-Fock state are tractable to classical computation, these results represent a first step towards implementing the QCM method in a quantum chemical trial circuit. Greater emphasis on more efficient representations of the Hamiltonian and classical preprocessing steps may enable the solution of larger systems on near-term quantum processors.
Quantum computers hold promise to circumvent the limitations of conventional computing for difficult molecular problems. However, the accumulation of quantum logic errors on real devices represents a major challenge, particularly in the pursuit of chemical accuracy requiring the inclusion of dynamical effects. In this work we implement the quantum computed moments (QCM) approach for hydrogen chain molecular systems up to H6_66​. On a superconducting quantum processor, Hamiltonian moments, ⟨Hp⟩\langle \mathcal{H}^p\rangle⟨Hp⟩ are computed with respect to the Hartree-Fock state, which are then employed in Lanczos expansion theory to determine an estimate for the ground-state energy which incorporates electronic correlations and manifestly improves on the variational result. Post-processing purification of the raw QCM data takes the estimate through the Hartree-Fock variational limit to within 99.9% of the exact electronic ground-state energy for the largest system studied, H6_66​. Calculated dissociation curves indicate precision at about 10mH for this system and as low as 0.1mH for molecular hydrogen, H2_22​, over a range of bond lengths. In the context of stringent precision requirements for chemical problems, these results provide strong evidence for the error suppression capability of the QCM method, particularly when coupled with post-processing error mitigation. Greater emphasis on more efficient representations of the Hamiltonian and classical preprocessing steps may enable the solution of larger systems on near-term quantum processors.
Quantum error mitigation (QEM) is essential for extracting chemically accurate results from near-term quantum hardware. Many widely used QEM methods rely on uncontrolled heuristics whose bias depends on the specific circuit and noise realization. In this work, we employ QESEM---a characterization-based, unbiased quasi-probabilistic mitigation method---on IBM's Aachen quantum processor to compute the ground-state potential energy surface (PES) of the symmetrically-stretched water molecule. We consider a classically-optimized, single-layer perfect-pairing tiled unitary product state ansatz. We map this ansatz to an 8-qubit register corresponding to a (4,4) active space and the STO-3G basis set. Compared to the statevector reference, we find that raw QPU results typically overestimate the ground-state energy by around 500~mHa across the scanned geometries. On the other hand, QESEM-mitigated results fall approximately within 100~mHa, 30~mHa, or ``chemical accuracy'' ($\sim$1.5~mHa), depending on the target precision. We benchmark QESEM at both loose (0.1~Ha) and tight (0.01~Ha) precision targets, evaluating both individual and merged batches of runs. As the precision target is tightened, the accuracy improves systematically, matching or exceeding results reported in the literature. We further quantify the sampling cost of these results, reporting the number of shots required at each precision level. Our results show how, with the current levels of hardware error, reaching the highest accuracies demands substantial QPU time. The results demonstrate that the characterization-based, unbiased error mitigation provided by QESEM allows to measure quantitatively meaningful potential energy surfaces on current quantum hardware. Concurrently, they highlight the sampling overhead associated with QEM, which remains a central bottleneck en route to larger chemical problems and higher precision.
With rapid progress being made in the development of platforms for quantum computation, there has been considerable interest in whether present-day and near-term devices can be used to solve problems of relevance. A commonly cited application area is the domain of quantum chemistry. While most experimental demonstrations of quantum chemical calculations on quantum devices have focused on the ground-state electronic energy of the system, other properties of the ground state, such as the electric dipole moment, are also of interest. Here, we employ the quantum-computed-moments method, based on the Lanczos cluster expansion, to estimate the dipole moment of the water molecule on an IBM Quantum superconducting quantum device. The noise-mitigated results agree with full-configuration-interaction calculations to within 0.03 ± 0.007 D ( 2 % ± 0.5 % ), compared with direct expectation-value determination (i.e., variational quantum eigensolver, VQE) with errors of the order of 0.07 D (5%), even when the VQE calculation is performed without noise. This demonstrates that moments-based energy-estimation techniques can be adapted to noise-robust evaluation of nonenergetic ground-state properties of chemical systems. We emphasize that the dipole moment is chosen as an example of such a property, but the method applies to any property determined by one- or two-body interactions.
Quantum computers have the potential to efficiently simulate large-scale quantum systems for which classical approaches are bound to fail. Even though several existing quantum devices now feature total qubit numbers of more than 100, their applicability remains plagued by the presence of noise and errors. Thus, the degree to which large quantum systems can successfully be simulated on these devices remains unclear. Here, we report on numerical results of physics-motivated variational ansatzes, as well as cloud simulations performed on several of IBM's superconducting quantum computers to simulate ground states of spin chains having a wide range of system sizes up to 102 qubits. Our numerical analysis shows that the accuracy of the ground-state energy and fidelity improves substantially by increasing the number of layers used in the ansatzes. From the cloud experiments, we find that the ground-state energies extracted from realizations across different quantum computers and system sizes reach the expected values to within errors that are small (i.e., on the percent level), including the inference of the energy density in the thermodynamic limit from these values. We achieve this accuracy through a combination of physics-motivated variational ansatzes, and efficient, scalable energy-measurement and error-mitigation protocols, including the use of a reference state in the zero-noise extrapolation. By using a 102-qubit system, we have been able to successfully apply up to 3186 CNOT gates in a single circuit when performing gate-error mitigation. Our accurate, error-mitigated results for random parameters in the ansatz states suggest that a standalone hybrid quantum-classical variational approach for large-scale XXZXXZXXZ models considered in this work is feasible.
In the lead up to fault tolerance, the utility of quantum computing will be determined by how adequately the effects of noise can be circumvented in quantum algorithms. Hybrid quantum-classical algorithms such as the variational quantum eigensolver (VQE) have been designed for the short-term regime. However, as problems scale, VQE results are generally scrambled by noise on present-day hardware. While error mitigation techniques alleviate these issues to some extent, there is a pressing need to develop algorithmic approaches with higher robustness to noise. Here, we explore the robustness properties of the recently introduced quantum computed moments (QCM) approach to ground state energy problems, and show through an analytic example how the underlying energy estimate explicitly filters out incoherent noise. Motivated by this observation, we implement QCM for a model of quantum magnetism on IBM Quantum hardware to examine the noise-filtering effect with increasing circuit depth. We find that QCM maintains a remarkably high degree of error robustness where VQE completely fails. On instances of the quantum magnetism model up to 20 qubits for ultra-deep trial state circuits of up to 500 CNOTs, QCM is still able to extract reasonable energy estimates. The observation is bolstered by an extensive set of experimental results. To match these results, VQE would need hardware improvement by some 2 orders of magnitude on error rates.