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:2309.09914 · Journal of Chemical Theory and Computation

Quantum Algorithm for Imaginary-Time Green’s Functions

Diksha Dhawan, Dominika Zgid, Mário Motta

Published May 18, 2024• 6 citations
arXivHTMLPDF

Abstract

Green’s function methods lead to ab initio, systematically improvable simulations of molecules and materials while providing access to multiple experimentally observable properties such as the density of states and the spectral function. The calculation of the exact one-particle Green’s function remains a significant challenge for classical computers and was attempted only on very small systems. Here, we present a hybrid quantum-classical algorithm to calculate the imaginary-time one-particle Green’s function. The proposed algorithm combines the variational quantum eigensolver and the quantum subspace expansion methods to calculate Green’s function in Lehmann’s representation. We demonstrate the validity of this algorithm by simulating H 2 and H 4 on quantum simulators and on IBM’s quantum devices.

DOI
10.1021/acs.jctc.4c00241

Related papers

Hybrid quantum-classical algorithm for computing imaginary-time correlation functions

Rihito Sakurai, Wataru Mizukami, Hiroshi Shinaoka

Quantitative descriptions of strongly correlated materials pose a considerable challenge in condensed matter physics and chemistry. A promising approach to address this problem is quantum embedding methods. In particular, the dynamical mean-field theory (DMFT) maps the original system to an effective quantum impurity model comprising correlated orbitals embedded in an electron bath. The biggest bottleneck in DMFT calculations is numerically solving the quantum impurity model, i.e., computing the Green's function. Past studies have proposed theoretical methods to compute the Green's function of a quantum impurity model in polynomial time using a quantum computer. So far, however, efficient methods for computing the imaginary-time Green's functions have not been established despite the advantages of the imaginary-time formulation. We propose a quantum-classical hybrid algorithm for computing imaginary-time Green's functions on quantum devices with limited hardware resources by applying the variational quantum simulation. Using a quantum circuit simulator, we verified this algorithm by computing Green's functions for a dimer model as well as a four-site impurity model obtained by DMFT calculations of the single-band Hubbard model, although our method can be applied to general imaginary-time correlation functions.

arXiv:2112.02764•Physical Review Research•Jun 16, 2022•30 citations

Quantum simulations of Green's functions for small superfluid systems

Samuel Aychet-Claisse, Denis Lacroix A, V. Somà +1 more

An end-to-end strategy for hybrid quantum-classical computations of Green's functions in many-body systems is presented and applied to the pairing model. The scheme makes explicit use of the spectral representation of the Green's function, which entails the calculation of the $N$-body ground state as well as eigenstates and associated energies of the $(N\pm1)$-body neighbors. While the former is accessed via variational techniques, the latter are constructed by means of the quantum subspace expansion method. Different ansatzes for the ground-state wave function, originating from either classical or quantum approaches, are tested and compared to exact calculations. The resulting one-body Green's functions prove to be accurate approximations of the exact one for a large range of parameters, including across the normal-to-superfluid transition. As a byproduct, this approach yields a good description of odd systems provided that the starting even system is well reproduced by the variational ansatz.

arXiv:2509.02272•Physical Review C•Apr 1, 2026•2 citations

Variational quantum eigensolver for dynamic correlation functions

Hongxiang Chen, Max Nusspickel, Jules Tilly +1 more

Recent practical approaches for the use of current generation noisy quantum devices in the simulation of quantum many-body problems have been dominated by the use of a variational quantum eigensolver (VQE). These coupled quantum-classical algorithms leverage the ability to perform many repeated measurements to avoid the currently prohibitive gate depths often required for exact quantum algorithms, with the restriction of a parametrized circuit to describe the states of interest. In this work, we show how the calculation of zero-temperature dynamic correlation functions defining the linear response characteristics of quantum systems can also be recast into a modified VQE algorithm, which can be incorporated into the current variational quantum infrastructure. This allows for these important physical expectation values describing the dynamics of the system to be directly converged on the frequency axis, and they approach exactness over all frequencies as the flexibility of the parametrization increases. The frequency resolution hence does not explicitly scale with gate depth, which is approximately twice as deep as a ground-state VQE. We apply the method to compute the single-particle Green's function of ab initio dihydrogen and lithium hydride molecules, and demonstrate the use of a practical active space embedding approach to extend to larger systems. While currently limited by the fidelity of two-qubit gates, whose number is increased compared to the ground-state algorithm on current devices, we believe the approach shows potential for the extraction of frequency dynamics of correlated systems on near-term quantum processors.

arXiv:2105.01703•Physical Review A•Sep 9, 2021•39 citations

Near-term quantum algorithm for computing molecular and materials properties based on recursive variational series methods

Phillip W. K. Jensen, Peter D. Johnson, Alexander A. Kunitsa

Determining the properties of molecules and materials is one of the premier applications of quantum computing. A major question in the field is: how might we use imperfect near-term quantum computers to solve problems of practical value? We propose a quantum algorithm to estimate the properties of molecules using near-term quantum devices. The method is a recursive variational series estimation method, where we expand an operator of interest in terms of Chebyshev polynomials and evaluate each term in the expansion using a variational quantum algorithm. We test our method by computing the one-particle Green's function in the energy domain and the autocorrelation function in the time domain.

arXiv:2206.09881•Physical Review A•Aug 25, 2023•4 citations

Occupation-number quantum-subspace-expansion approach to computing the single-particle Green function: An opportunity for noise filtering

B. Gauthier, Peter Rosenberg, Alexandre Foley +1 more

We introduce a hybrid quantum-classical algorithm to compute the Green function for strongly correlated electrons on noisy intermediate-scale quantum (NISQ) devices. The technique consists in the construction of a nonorthogonal excitation basis composed of a set of single-particle excitations generated by occupation-number operators. The excited sectors of the Hamiltonian in this basis can then be measured on the quantum device and a classical postprocessing procedure yields the Green function in the Lehmann representation. The technique allows for noise filtering, a useful feature for NISQ devices. To validate the approach, we carry out a set of proof-of-principle calculations on the single-band Hubbard model on IBM quantum hardware. For a two-site system we find good agreement between the results of quantum simulations and the exact result for the local spectral function. This proof of principle also shows that the noise filtering provides a reliable way to get rid of satellite peaks present in the spectral weight obtained from a NISQ device. A simulation of a four-site system carried out on classical hardware suggests that the approach can achieve similar accuracy for larger systems.

arXiv:2312.13497•Physical Review A•Sep 25, 2024•0 citations

Subspace-Search Quantum Imaginary Time Evolution for Excited State Computations

Cameron Cianci, Léa Ferreira dos Santos, Víctor S. Batista

Quantum systems in excited states are attracting significant interest with the advent of noisy intermediate-scale quantum (NISQ) devices. While ground states of small molecular systems are typically explored using hybrid variational algorithms like the variational quantum eigensolver (VQE), the study of excited states has received much less attention, partly due to the absence of efficient algorithms. In this work, we introduce the subspace search quantum imaginary time evolution (SSQITE) method, which calculates excited states using quantum devices by integrating key elements of the subspace search variational quantum eigensolver (SSVQE) and the variational quantum imaginary time evolution (VarQITE) method. The effectiveness of SSQITE is demonstrated through calculations of low-lying excited states of benchmark model systems including H 2 and LiH molecules. A toy Hamiltonian is also employed to demonstrate that the robustness of VarQITE in avoiding local minima extends to its use in excited state algorithms. With this robustness in avoiding local minima, SSQITE shows promise for advancing quantum computations of excited states across a wide range of applications.

arXiv:2407.11182•Journal of Chemical Theory and Computation•Oct 1, 2024•7 citations
Green’s function methods lead to ab initio, systematically improvable simulations of molecules and materials while providing access to multiple experimentally observable properties such as the density of states and the spectral function. The calculation of the exact one-particle Green’s function remains a significant challenge for classical computers and was attempted only on very small systems. Here, we present a hybrid quantum-classical algorithm to calculate the imaginary-time one-particle Green’s function. The proposed algorithm combines the variational quantum eigensolver and the quantum subspace expansion methods to calculate Green’s function in Lehmann’s representation. We demonstrate the validity of this algorithm by simulating H 2 and H 4 on quantum simulators and on IBM’s quantum devices.
Quantitative descriptions of strongly correlated materials pose a considerable challenge in condensed matter physics and chemistry. A promising approach to address this problem is quantum embedding methods. In particular, the dynamical mean-field theory (DMFT) maps the original system to an effective quantum impurity model comprising correlated orbitals embedded in an electron bath. The biggest bottleneck in DMFT calculations is numerically solving the quantum impurity model, i.e., computing the Green's function. Past studies have proposed theoretical methods to compute the Green's function of a quantum impurity model in polynomial time using a quantum computer. So far, however, efficient methods for computing the imaginary-time Green's functions have not been established despite the advantages of the imaginary-time formulation. We propose a quantum-classical hybrid algorithm for computing imaginary-time Green's functions on quantum devices with limited hardware resources by applying the variational quantum simulation. Using a quantum circuit simulator, we verified this algorithm by computing Green's functions for a dimer model as well as a four-site impurity model obtained by DMFT calculations of the single-band Hubbard model, although our method can be applied to general imaginary-time correlation functions.
An end-to-end strategy for hybrid quantum-classical computations of Green's functions in many-body systems is presented and applied to the pairing model. The scheme makes explicit use of the spectral representation of the Green's function, which entails the calculation of the NNN-body ground state as well as eigenstates and associated energies of the (N±1)(N\pm1)(N±1)-body neighbors. While the former is accessed via variational techniques, the latter are constructed by means of the quantum subspace expansion method. Different ansatzes for the ground-state wave function, originating from either classical or quantum approaches, are tested and compared to exact calculations. The resulting one-body Green's functions prove to be accurate approximations of the exact one for a large range of parameters, including across the normal-to-superfluid transition. As a byproduct, this approach yields a good description of odd systems provided that the starting even system is well reproduced by the variational ansatz.
Recent practical approaches for the use of current generation noisy quantum devices in the simulation of quantum many-body problems have been dominated by the use of a variational quantum eigensolver (VQE). These coupled quantum-classical algorithms leverage the ability to perform many repeated measurements to avoid the currently prohibitive gate depths often required for exact quantum algorithms, with the restriction of a parametrized circuit to describe the states of interest. In this work, we show how the calculation of zero-temperature dynamic correlation functions defining the linear response characteristics of quantum systems can also be recast into a modified VQE algorithm, which can be incorporated into the current variational quantum infrastructure. This allows for these important physical expectation values describing the dynamics of the system to be directly converged on the frequency axis, and they approach exactness over all frequencies as the flexibility of the parametrization increases. The frequency resolution hence does not explicitly scale with gate depth, which is approximately twice as deep as a ground-state VQE. We apply the method to compute the single-particle Green's function of ab initio dihydrogen and lithium hydride molecules, and demonstrate the use of a practical active space embedding approach to extend to larger systems. While currently limited by the fidelity of two-qubit gates, whose number is increased compared to the ground-state algorithm on current devices, we believe the approach shows potential for the extraction of frequency dynamics of correlated systems on near-term quantum processors.
Determining the properties of molecules and materials is one of the premier applications of quantum computing. A major question in the field is: how might we use imperfect near-term quantum computers to solve problems of practical value? We propose a quantum algorithm to estimate the properties of molecules using near-term quantum devices. The method is a recursive variational series estimation method, where we expand an operator of interest in terms of Chebyshev polynomials and evaluate each term in the expansion using a variational quantum algorithm. We test our method by computing the one-particle Green's function in the energy domain and the autocorrelation function in the time domain.
We introduce a hybrid quantum-classical algorithm to compute the Green function for strongly correlated electrons on noisy intermediate-scale quantum (NISQ) devices. The technique consists in the construction of a nonorthogonal excitation basis composed of a set of single-particle excitations generated by occupation-number operators. The excited sectors of the Hamiltonian in this basis can then be measured on the quantum device and a classical postprocessing procedure yields the Green function in the Lehmann representation. The technique allows for noise filtering, a useful feature for NISQ devices. To validate the approach, we carry out a set of proof-of-principle calculations on the single-band Hubbard model on IBM quantum hardware. For a two-site system we find good agreement between the results of quantum simulations and the exact result for the local spectral function. This proof of principle also shows that the noise filtering provides a reliable way to get rid of satellite peaks present in the spectral weight obtained from a NISQ device. A simulation of a four-site system carried out on classical hardware suggests that the approach can achieve similar accuracy for larger systems.
Quantum systems in excited states are attracting significant interest with the advent of noisy intermediate-scale quantum (NISQ) devices. While ground states of small molecular systems are typically explored using hybrid variational algorithms like the variational quantum eigensolver (VQE), the study of excited states has received much less attention, partly due to the absence of efficient algorithms. In this work, we introduce the subspace search quantum imaginary time evolution (SSQITE) method, which calculates excited states using quantum devices by integrating key elements of the subspace search variational quantum eigensolver (SSVQE) and the variational quantum imaginary time evolution (VarQITE) method. The effectiveness of SSQITE is demonstrated through calculations of low-lying excited states of benchmark model systems including H 2 and LiH molecules. A toy Hamiltonian is also employed to demonstrate that the robustness of VarQITE in avoiding local minima extends to its use in excited state algorithms. With this robustness in avoiding local minima, SSQITE shows promise for advancing quantum computations of excited states across a wide range of applications.