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:2206.09881 · Physical Review A

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

Published Aug 25, 2023• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.108.022422

Related papers

Quantum Computing for Chemistry: From Quantum Computing with Molecules to Algorithmic Design

Phillip W. K. Jensen

Determining properties of molecules and materials is one of the premier applications of quantum computing. A major question in the thesis is: how might we use long- and near-term quantum computers to solve problems of chemical interest? First, we give an overview of the standard model in electronic structure theory. Electronic structure methods have been improved significantly in the last decades; however, it is still a non-trivial task to fit experimental results for complex molecular systems. Second, we introduce quantum computation and quantum algorithms as an alternative device to solve the electronic structure problem. In the remaining part of the thesis, we describe the research projects. The thesis covers two (rather different) research directions: software and hardware projects. For the software part, we contribute with two new algorithms: Phase-Estimation Interval Target Energy Readout (PHILTER) and Recursive Variational Series Estimation (RVSE), where both are designed to be run on a quantum digital computer and extract materials and molecular properties. The PHILTER algorithm is a modification of the Quantum Phase Estimation and is designed to compute excited state energies within a target energy-interval for cases where good approximations for the states in the target energy-interval are either unknown or hard to prepare. For the RVSE algorithm, we pose the question: how might we use imperfect near-term quantum computers to solve problems of practical value? In this work, we introduce a method to compute expectation values of general functions of a Hamiltonian with respect to an input state using low depth quantum circuits suitable for noisy intermediate-scale quantum computers. For the hardware part, we propose a new chemical quantum computer by using molecular electronics as a basis for quantum computing. In this work, we pose the question: can scattering of electrons in molecules produce quantum gates? We construct one-qubit gates using one-electron scattering in molecules, and two-qubit controlled-phase gates using electron-electron scattering along metallic leads. We show initial applications of the framework by illustrating one-qubit gates using the molecular electronic structure of molecular hydrogen as a baseline model.

•Nov 1, 2022•0 citations

Variational Quantum Computation of Molecular Linear Response Properties on a Superconducting Quantum Processor

Kaixuan Huang, Xiaoxia Cai, Hao Li +12 more

Simulating response properties of molecules is crucial for interpreting experimental spectroscopies and accelerating materials design. However, it remains a long-standing computational challenge for electronic structure methods on classical computers. While quantum computers hold the promise of solving this problem more efficiently in the long run, existing quantum algorithms requiring deep quantum circuits are infeasible for near-term noisy quantum processors. Herein, we introduce a pragmatic variational quantum response (VQR) algorithm for response properties, which circumvents the need for deep quantum circuits. Using this algorithm, we report the first simulation of linear response properties of molecules including dynamic polarizabilities and absorption spectra on a superconducting quantum processor. Our results indicate that a large class of important dynamical properties, such as Green's functions, are within the reach of near-term quantum hardware using this algorithm in combination with suitable error mitigation techniques.

arXiv:2201.02426•The Journal of Physical Chemistry Letters•Sep 26, 2022•49 citations

Quantum Algorithm for Imaginary-Time Green’s Functions

Diksha Dhawan, Dominika Zgid, Mário Motta

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.

arXiv:2309.09914•Journal of Chemical Theory and Computation•May 18, 2024•6 citations

Computing Green's functions and improving ground state energy estimation on quantum computers with Liouvillian recursion

Jérôme Leblanc, Olivier Nahman-Lévesque, Julien Forget +3 more

We present a quantum-classical hybrid implementation of the Liouvillian recursion method to compute many-body Green's functions using a quantum computer. From an approximate ground state preparation circuit, this algorithm produces the local ($r=r'$) and inter-site ($r\neq r'$) Green's functions $G_{rr'}(ω)$ by measuring observables generated recursively. We demonstrate the approach on a superconducting quantum processor for the open-boundary four-site Hubbard model. We then use the computed Green's functions as input to the Galitskii-Migdal formula to produce better ground state energy estimation than the expectation value of the Hamiltonian for the approximate circuit. Empirical results indicate exponential convergence in the number of iterations, yielding a computational complexity polynomial in the Green's-function accuracy, as measured with the Wasserstein distance. Our results also indicate significant robustness to noise and to inaccuracies of the ground state preparation, providing evidence that Liouvillian recursion is well adapted to the constraints of near-term quantum computing.

quant-pharXiv:2603.05349•Mar 6, 2026•0 citations

Quantum Equation of Motion with Orbital Optimization for Computing Molecular Properties in Near-Term Quantum Computing

Phillip W. K. Jensen, Erik Rosendahl Kjellgren, Peter Reinholdt +4 more

Determining the properties of molecules and materials is one of the premier applications of quantum computing. A major question in the field is how to use imperfect near-term quantum computers to solve problems of practical value. Inspired by the recently developed variants of the quantum counterpart of the equation-of-motion (qEOM) approach and the orbital-optimized variational quantum eigensolver (oo-VQE), we present a quantum algorithm (oo-VQE-qEOM) for the calculation of molecular properties by computing expectation values on a quantum computer. We perform noise-free quantum simulations of BeH 2 in the series of STO-3G/6-31G/6-31G* basis sets and of H 4 and H 2 O in 6-31G using an active space of four electrons and four spatial orbitals (8 qubits) to evaluate excitation energies, electronic absorption, and, for twisted H 4, circular dichroism spectra. We demonstrate that the proposed algorithm can reproduce the results of conventional classical CASSCF calculations for these molecular systems.

arXiv:2312.12386•Journal of Chemical Theory and Computation•May 3, 2024•18 citations

Computing Ground State Properties with Early Fault-Tolerant Quantum Computers

Ruizhe Zhang, Guoming Wang, Peter D. Johnson

Significant effort in applied quantum computing has been devoted to the problem of ground state energy estimation for molecules and materials. Yet, for many applications of practical value, additional properties of the ground state must be estimated. These include Green's functions used to compute electron transport in materials and the one-particle reduced density matrices used to compute electric dipoles of molecules. In this paper, we propose a quantum-classical hybrid algorithm to efficiently estimate such ground state properties with high accuracy using low-depth quantum circuits. We provide an analysis of various costs (circuit repetitions, maximal evolution time, and expected total runtime) as a function of target accuracy, spectral gap, and initial ground state overlap. This algorithm suggests a concrete approach to using early fault tolerant quantum computers for carrying out industry-relevant molecular and materials calculations.

arXiv:2109.13957•Quantum•Jul 11, 2022•47 citations
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.
Determining properties of molecules and materials is one of the premier applications of quantum computing. A major question in the thesis is: how might we use long- and near-term quantum computers to solve problems of chemical interest? First, we give an overview of the standard model in electronic structure theory. Electronic structure methods have been improved significantly in the last decades; however, it is still a non-trivial task to fit experimental results for complex molecular systems. Second, we introduce quantum computation and quantum algorithms as an alternative device to solve the electronic structure problem. In the remaining part of the thesis, we describe the research projects. The thesis covers two (rather different) research directions: software and hardware projects. For the software part, we contribute with two new algorithms: Phase-Estimation Interval Target Energy Readout (PHILTER) and Recursive Variational Series Estimation (RVSE), where both are designed to be run on a quantum digital computer and extract materials and molecular properties. The PHILTER algorithm is a modification of the Quantum Phase Estimation and is designed to compute excited state energies within a target energy-interval for cases where good approximations for the states in the target energy-interval are either unknown or hard to prepare. For the RVSE algorithm, we pose the question: how might we use imperfect near-term quantum computers to solve problems of practical value? In this work, we introduce a method to compute expectation values of general functions of a Hamiltonian with respect to an input state using low depth quantum circuits suitable for noisy intermediate-scale quantum computers. For the hardware part, we propose a new chemical quantum computer by using molecular electronics as a basis for quantum computing. In this work, we pose the question: can scattering of electrons in molecules produce quantum gates? We construct one-qubit gates using one-electron scattering in molecules, and two-qubit controlled-phase gates using electron-electron scattering along metallic leads. We show initial applications of the framework by illustrating one-qubit gates using the molecular electronic structure of molecular hydrogen as a baseline model.
Simulating response properties of molecules is crucial for interpreting experimental spectroscopies and accelerating materials design. However, it remains a long-standing computational challenge for electronic structure methods on classical computers. While quantum computers hold the promise of solving this problem more efficiently in the long run, existing quantum algorithms requiring deep quantum circuits are infeasible for near-term noisy quantum processors. Herein, we introduce a pragmatic variational quantum response (VQR) algorithm for response properties, which circumvents the need for deep quantum circuits. Using this algorithm, we report the first simulation of linear response properties of molecules including dynamic polarizabilities and absorption spectra on a superconducting quantum processor. Our results indicate that a large class of important dynamical properties, such as Green's functions, are within the reach of near-term quantum hardware using this algorithm in combination with suitable error mitigation techniques.
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.
We present a quantum-classical hybrid implementation of the Liouvillian recursion method to compute many-body Green's functions using a quantum computer. From an approximate ground state preparation circuit, this algorithm produces the local (r=r′r=r'r=r′) and inter-site (r≠r′r\neq r'r=r′) Green's functions Grr′(ω)G_{rr'}(ω)Grr′​(ω) by measuring observables generated recursively. We demonstrate the approach on a superconducting quantum processor for the open-boundary four-site Hubbard model. We then use the computed Green's functions as input to the Galitskii-Migdal formula to produce better ground state energy estimation than the expectation value of the Hamiltonian for the approximate circuit. Empirical results indicate exponential convergence in the number of iterations, yielding a computational complexity polynomial in the Green's-function accuracy, as measured with the Wasserstein distance. Our results also indicate significant robustness to noise and to inaccuracies of the ground state preparation, providing evidence that Liouvillian recursion is well adapted to the constraints of near-term quantum computing.
Determining the properties of molecules and materials is one of the premier applications of quantum computing. A major question in the field is how to use imperfect near-term quantum computers to solve problems of practical value. Inspired by the recently developed variants of the quantum counterpart of the equation-of-motion (qEOM) approach and the orbital-optimized variational quantum eigensolver (oo-VQE), we present a quantum algorithm (oo-VQE-qEOM) for the calculation of molecular properties by computing expectation values on a quantum computer. We perform noise-free quantum simulations of BeH 2 in the series of STO-3G/6-31G/6-31G* basis sets and of H 4 and H 2 O in 6-31G using an active space of four electrons and four spatial orbitals (8 qubits) to evaluate excitation energies, electronic absorption, and, for twisted H 4, circular dichroism spectra. We demonstrate that the proposed algorithm can reproduce the results of conventional classical CASSCF calculations for these molecular systems.
Significant effort in applied quantum computing has been devoted to the problem of ground state energy estimation for molecules and materials. Yet, for many applications of practical value, additional properties of the ground state must be estimated. These include Green's functions used to compute electron transport in materials and the one-particle reduced density matrices used to compute electric dipoles of molecules. In this paper, we propose a quantum-classical hybrid algorithm to efficiently estimate such ground state properties with high accuracy using low-depth quantum circuits. We provide an analysis of various costs (circuit repetitions, maximal evolution time, and expected total runtime) as a function of target accuracy, spectral gap, and initial ground state overlap. This algorithm suggests a concrete approach to using early fault tolerant quantum computers for carrying out industry-relevant molecular and materials calculations.