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:2308.12524 · International Conference on Quantum Technologies, Communications, Computing, Hardware and Embedded Systems Security (iQ-CCHESS)

On the Study of Partial Qubit Hamiltonian for Efficient Molecular Simulation Using Variational Quantum Eigensolvers

Harshdeep Singh, Sabyashachi Mishra, Sonjoy Majumder

Published Sep 15, 2023• 1 citation
arXivHTMLPDF

Abstract

Quantum computing is being extensively used in quantum chemistry, especially in simulating simple molecules and evaluating properties like the ground state energy, dipole moment, etc. The transformation of a molecular Hamiltonian from the fermionic space to the qubit space provides us with a series of Pauli strings and the energy calculation involves the evaluation of the expectation values of all these individual strings. This introduces a major bottleneck for applications of VQEs in quantum chemistry. Unlike the fermionic Hamiltonian, the terms in a qubit Hamiltonian are additive and the present paper exploits this property to describe a new approach for extracting information from the partial qubit Hamiltonian of simple molecules to design more efficient variational quantum eigensolvers. In the partial (qubit) Hamiltonian approach (PHA), the qubit Hamiltonian is studied term-by-term to understand their relative contributions to the overall energy, and a partial Hamiltonian is constructed with fewer Pauli strings that can resolve the entire Hamiltonian. With PHA, we can simulate molecules at a much lower computational cost with a truncated Hamiltonian. Additionally, the outcomes of the measurements with PHA quench the error due to noise introduced by the quantum circuits. We have also demonstrated the application of PHA as an initialization technique, where the simple partial Hamiltonian can be used to find an optimal starting point for a more complex system.

DOI
10.1109/iq-cchess56596.2023.10391502

Related papers

SHARC-VQE: Simplified Hamiltonian approach with refinement and correction enabled variational quantum eigensolver for molecular simulation

Harshdeep Singh, Sonjoy Majumder, Sabyashachi Mishra

Quantum computing is finding increasingly more applications in quantum chemistry, particularly to simulate electronic structure and molecular properties of simple systems. The transformation of a molecular Hamiltonian from the fermionic space to the qubit space results in a series of Pauli strings. Calculating the energy then involves evaluating the expectation values of each of these strings, which presents a significant bottleneck for applying variational quantum eigensolvers (VQEs) in quantum chemistry. Unlike fermionic Hamiltonians, the terms in a qubit Hamiltonian are additive. This work leverages this property to introduce a novel method for extracting information from the partial qubit Hamiltonian, thereby enhancing the efficiency of VQEs. This work introduces the SHARC-VQE (Simplified Hamiltonian Approximation, Refinement, and Correction-VQE) method, where the full molecular Hamiltonian is partitioned into two parts based on the ease of quantum execution. The easy-to-execute part constitutes the partial Hamiltonian, and the remaining part, while more complex to execute, is generally less significant. The latter is approximated by a refined operator and added up as a correction into the partial Hamiltonian. SHARC-VQE significantly reduces computational costs for molecular simulations. The cost of a single energy measurement can be reduced from O(N4ϵ2) to O(1ϵ2) for a system of N qubits and accuracy ϵ, while the overall cost of VQE can be reduced from O(N7ϵ2) to O(N3ϵ2). Furthermore, measurement outcomes using SHARC-VQE are less prone to errors induced by noise from quantum circuits, reducing the errors from 20%-40% to 5%-10% without any additional error correction or mitigation technique. In addition, the SHARC-VQE is demonstrated as an initialization technique, where the simplified partial Hamiltonian is used to identify an optimal starting point for a complex problem. Overall, this method improves the efficiency of VQEs and enhances the accuracy and reliability of quantum simulations by mitigating noise and overcoming computational challenges.

arXiv:2407.12305•The Journal of Chemical Physics•Mar 19, 2025•7 citations

Extracting Information from Partial Hamiltonian of Simple Molecules for Designing Efficient Variational Quantum Eigensolvers

Harshdeep Singh, Sabyashachi Mishra, Sonjoy Majumder

This work aims to study the qubit Hamiltonian of a molecule term-by-term to understand their contributions, and if any information can be extracted by just considering a few selected terms of the Hamiltonian, a technique called partial Hamiltonian analysis (PHA).

Quantum 2.0 Conference and Exhibition•Jan 1, 2023•0 citations

Improving the Accuracy of Variational Quantum Eigensolvers with Fewer Qubits Using Orbital Optimization

Joel Bierman, Yingzhou Li, Jianfeng Lu

Near-term quantum computers will be limited in the number of qubits on which they can process information as well as the depth of the circuits that they can coherently carry out. To date, experimental demonstrations of algorithms such as the Variational Quantum Eigensolver (VQE) have been limited to small molecules using minimal basis sets for this reason. In this work we propose incorporating an orbital optimization scheme into quantum eigensolvers wherein a parametrized partial unitary transformation is applied to the basis functions set in order to reduce the number of qubits required for a given problem. The optimal transformation is found by minimizing the ground state energy with respect to this partial unitary matrix. Through numerical simulations of small molecules up to 16 spin orbitals, we demonstrate that this method has the ability to greatly extend the capabilities of near-term quantum computers with regard to the electronic structure problem. We find that VQE paired with orbital optimization consistently achieves lower ground state energies than traditional VQE when using the same number of qubits and even frequently achieves lower ground state energies than VQE methods using more qubits.

arXiv:2208.14431•Journal of Chemical Theory and Computation•Jan 25, 2023•24 citations

Truncation technique for variational quantum eigensolver for Molecular Hamiltonians

Xu, Qidong, Kanav Setia

The variational quantum eigensolver (VQE) is one of the most promising quantum algorithms for the near-term noisy intermediate-scale quantum (NISQ) devices. The VQE typically involves finding the minimum energy of a quantum Hamiltonian through classical optimization of a parametrized quantum ansatz. One of the bottlenecks in VQEs is the number of quantum circuits to be measured. In this work, we propose a physically intuitive truncation technique that starts the optimization procedure with a truncated Hamiltonian and then gradually transitions to the optimization for the original Hamiltonian via an operator classification method. This strategy allows us to reduce the required number of evaluations for the expectation value of Hamiltonian on a quantum computer. The reduction in required quantum resources for our strategy is substantial and likely scales with the system size. With numerical simulations, we demonstrate our method for various molecular systems.

arXiv:2402.01630•Feb 2, 2024•0 citations

Noise-Mitigated Variational Quantum Eigensolver with Pre-training and Zero-Noise Extrapolation

Wanqi Sun, Jungang Xu, Chenghua Duan

As a hybrid quantum-classical algorithm, the variational quantum eigensolver is widely applied in quantum chemistry simulations, especially in computing the electronic structure of complex molecular systems. However, on existing noisy intermediate-scale quantum devices, some factors such as quantum decoherence, measurement errors, and gate operation imprecisions are unavoidable. To overcome these challenges, this study proposes an efficient noise-mitigating variational quantum eigensolver for accurate computation of molecular ground state energies in noisy environments. We design the quantum circuit with reference to the structure of matrix product states and utilize it to pre-train the circuit parameters, which ensures circuit stability and mitigates fluctuations caused by initialization. We also employ zero-noise extrapolation to mitigate quantum noise and combine it with neural networks to improve the accuracy of the noise-fitting function, which significantly eliminates noise interference. Furthermore, we implement an intelligent grouping strategy for measuring Hamiltonian Pauli strings, which not only reduces measurement errors but also improves sampling efficiency. We perform numerical simulations to solve the ground state energy of the H4molecule by using MindSpore Quantum framework, and the results demonstrate that our algorithm can constrain noise errors within the range of $\mathcal{O}\left( {{{10}^{ - 2}}} \right)\sim \mathcal{O}\left( {{{10}^{ - 1}}} \right)$, outperforming mainstream variational quantum eigensolvers. This work provides a new strategy for high-precision quantum chemistry calculations on near-term noisy quantum hardware.

arXiv:2501.01646•IEEE International Conference on Acoustics Speech and Signal Processing•Mar 12, 2025•1 citation

Folded Spectrum VQE: A Quantum Computing Method for the Calculation of Molecular Excited States

Lila Cadi Tazi, Alex J. W. Thom

High Resolution Image Download MS PowerPoint Slide The recent developments of quantum computing present novel potential pathways for quantum chemistry as the scaling of the computational power of quantum computers could be harnessed to naturally encode and solve electronic structure problems. Theoretically exact quantum algorithms for chemistry have been proposed (e.g., quantum phase estimation), but the limited capabilities of current noisy intermediate-scale quantum devices motivated the development of less demanding hybrid algorithms. In this context, the variational quantum eigensolver (VQE) algorithm was successfully introduced as an effective method to compute the ground-state energies of small molecules. This study investigates the folded spectrum (FS) method as an extension of the VQE algorithm for the computation of molecular excited states. It provides the possibility of directly computing excited states around a selected target energy using the same or quantum circuit as for the ground-state calculation. Inspired by the variance-based methods from the quantum Monte Carlo literature, the FS method minimizes the energy variance, thus, in principle, requiring a computationally expensive squared Hamiltonian to be applied. We alleviate this potentially poor scaling by employing a Pauli grouping procedure to identify sets of commuting Pauli strings that can be evaluated simultaneously. This allows for a significant reduction in the computational cost. We applied the FS-VQE method to small molecules (H 2, LiH), obtaining all electronic excited states with chemical accuracy on ideal quantum simulators. Furthermore, we explore the application of quantum error mitigation techniques, demonstrating improved energy accuracy on noisy simulators compared with simulations without mitigation.

arXiv:2305.04783•Journal of Chemical Theory and Computation•Mar 16, 2024•29 citations
Quantum computing is being extensively used in quantum chemistry, especially in simulating simple molecules and evaluating properties like the ground state energy, dipole moment, etc. The transformation of a molecular Hamiltonian from the fermionic space to the qubit space provides us with a series of Pauli strings and the energy calculation involves the evaluation of the expectation values of all these individual strings. This introduces a major bottleneck for applications of VQEs in quantum chemistry. Unlike the fermionic Hamiltonian, the terms in a qubit Hamiltonian are additive and the present paper exploits this property to describe a new approach for extracting information from the partial qubit Hamiltonian of simple molecules to design more efficient variational quantum eigensolvers. In the partial (qubit) Hamiltonian approach (PHA), the qubit Hamiltonian is studied term-by-term to understand their relative contributions to the overall energy, and a partial Hamiltonian is constructed with fewer Pauli strings that can resolve the entire Hamiltonian. With PHA, we can simulate molecules at a much lower computational cost with a truncated Hamiltonian. Additionally, the outcomes of the measurements with PHA quench the error due to noise introduced by the quantum circuits. We have also demonstrated the application of PHA as an initialization technique, where the simple partial Hamiltonian can be used to find an optimal starting point for a more complex system.
Quantum computing is finding increasingly more applications in quantum chemistry, particularly to simulate electronic structure and molecular properties of simple systems. The transformation of a molecular Hamiltonian from the fermionic space to the qubit space results in a series of Pauli strings. Calculating the energy then involves evaluating the expectation values of each of these strings, which presents a significant bottleneck for applying variational quantum eigensolvers (VQEs) in quantum chemistry. Unlike fermionic Hamiltonians, the terms in a qubit Hamiltonian are additive. This work leverages this property to introduce a novel method for extracting information from the partial qubit Hamiltonian, thereby enhancing the efficiency of VQEs. This work introduces the SHARC-VQE (Simplified Hamiltonian Approximation, Refinement, and Correction-VQE) method, where the full molecular Hamiltonian is partitioned into two parts based on the ease of quantum execution. The easy-to-execute part constitutes the partial Hamiltonian, and the remaining part, while more complex to execute, is generally less significant. The latter is approximated by a refined operator and added up as a correction into the partial Hamiltonian. SHARC-VQE significantly reduces computational costs for molecular simulations. The cost of a single energy measurement can be reduced from O(N4ϵ2) to O(1ϵ2) for a system of N qubits and accuracy ϵ, while the overall cost of VQE can be reduced from O(N7ϵ2) to O(N3ϵ2). Furthermore, measurement outcomes using SHARC-VQE are less prone to errors induced by noise from quantum circuits, reducing the errors from 20%-40% to 5%-10% without any additional error correction or mitigation technique. In addition, the SHARC-VQE is demonstrated as an initialization technique, where the simplified partial Hamiltonian is used to identify an optimal starting point for a complex problem. Overall, this method improves the efficiency of VQEs and enhances the accuracy and reliability of quantum simulations by mitigating noise and overcoming computational challenges.
Near-term quantum computers will be limited in the number of qubits on which they can process information as well as the depth of the circuits that they can coherently carry out. To date, experimental demonstrations of algorithms such as the Variational Quantum Eigensolver (VQE) have been limited to small molecules using minimal basis sets for this reason. In this work we propose incorporating an orbital optimization scheme into quantum eigensolvers wherein a parametrized partial unitary transformation is applied to the basis functions set in order to reduce the number of qubits required for a given problem. The optimal transformation is found by minimizing the ground state energy with respect to this partial unitary matrix. Through numerical simulations of small molecules up to 16 spin orbitals, we demonstrate that this method has the ability to greatly extend the capabilities of near-term quantum computers with regard to the electronic structure problem. We find that VQE paired with orbital optimization consistently achieves lower ground state energies than traditional VQE when using the same number of qubits and even frequently achieves lower ground state energies than VQE methods using more qubits.
The variational quantum eigensolver (VQE) is one of the most promising quantum algorithms for the near-term noisy intermediate-scale quantum (NISQ) devices. The VQE typically involves finding the minimum energy of a quantum Hamiltonian through classical optimization of a parametrized quantum ansatz. One of the bottlenecks in VQEs is the number of quantum circuits to be measured. In this work, we propose a physically intuitive truncation technique that starts the optimization procedure with a truncated Hamiltonian and then gradually transitions to the optimization for the original Hamiltonian via an operator classification method. This strategy allows us to reduce the required number of evaluations for the expectation value of Hamiltonian on a quantum computer. The reduction in required quantum resources for our strategy is substantial and likely scales with the system size. With numerical simulations, we demonstrate our method for various molecular systems.
As a hybrid quantum-classical algorithm, the variational quantum eigensolver is widely applied in quantum chemistry simulations, especially in computing the electronic structure of complex molecular systems. However, on existing noisy intermediate-scale quantum devices, some factors such as quantum decoherence, measurement errors, and gate operation imprecisions are unavoidable. To overcome these challenges, this study proposes an efficient noise-mitigating variational quantum eigensolver for accurate computation of molecular ground state energies in noisy environments. We design the quantum circuit with reference to the structure of matrix product states and utilize it to pre-train the circuit parameters, which ensures circuit stability and mitigates fluctuations caused by initialization. We also employ zero-noise extrapolation to mitigate quantum noise and combine it with neural networks to improve the accuracy of the noise-fitting function, which significantly eliminates noise interference. Furthermore, we implement an intelligent grouping strategy for measuring Hamiltonian Pauli strings, which not only reduces measurement errors but also improves sampling efficiency. We perform numerical simulations to solve the ground state energy of the H4molecule by using MindSpore Quantum framework, and the results demonstrate that our algorithm can constrain noise errors within the range of O(10−2)∼O(10−1)\mathcal{O}\left( {{{10}^{ - 2}}} \right)\sim \mathcal{O}\left( {{{10}^{ - 1}}} \right)O(10−2)∼O(10−1), outperforming mainstream variational quantum eigensolvers. This work provides a new strategy for high-precision quantum chemistry calculations on near-term noisy quantum hardware.
High Resolution Image Download MS PowerPoint Slide The recent developments of quantum computing present novel potential pathways for quantum chemistry as the scaling of the computational power of quantum computers could be harnessed to naturally encode and solve electronic structure problems. Theoretically exact quantum algorithms for chemistry have been proposed (e.g., quantum phase estimation), but the limited capabilities of current noisy intermediate-scale quantum devices motivated the development of less demanding hybrid algorithms. In this context, the variational quantum eigensolver (VQE) algorithm was successfully introduced as an effective method to compute the ground-state energies of small molecules. This study investigates the folded spectrum (FS) method as an extension of the VQE algorithm for the computation of molecular excited states. It provides the possibility of directly computing excited states around a selected target energy using the same or quantum circuit as for the ground-state calculation. Inspired by the variance-based methods from the quantum Monte Carlo literature, the FS method minimizes the energy variance, thus, in principle, requiring a computationally expensive squared Hamiltonian to be applied. We alleviate this potentially poor scaling by employing a Pauli grouping procedure to identify sets of commuting Pauli strings that can be evaluated simultaneously. This allows for a significant reduction in the computational cost. We applied the FS-VQE method to small molecules (H 2, LiH), obtaining all electronic excited states with chemical accuracy on ideal quantum simulators. Furthermore, we explore the application of quantum error mitigation techniques, demonstrating improved energy accuracy on noisy simulators compared with simulations without mitigation.