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:1905.03742 · npj Quantum Information

Calculating energy derivatives for quantum chemistry on a quantum computer

Thomas E. O’Brien, Bruno Senjean, Ramiro Sagastizabal, Xavier Bonet-Monroig, Alicja Dutkiewicz, Francesco Buda, L. DiCarlo, Lucas Visscher

Published Dec 11, 2019• 101 citations
arXivHTMLPDF

Abstract

Abstract Modeling chemical reactions and complicated molecular systems has been proposed as the “killer application” of a future quantum computer. Accurate calculations of derivatives of molecular eigenenergies are essential toward this end, allowing for geometry optimization, transition state searches, predictions of the response to an applied electric or magnetic field, and molecular dynamics simulations. In this work, we survey methods to calculate energy derivatives, and present two new methods: one based on quantum phase estimation, the other on a low-order response approximation. We calculate asymptotic error bounds and approximate computational scalings for the methods presented. Implementing these methods, we perform geometry optimization on an experimental quantum processor, estimating the equilibrium bond length of the dihydrogen molecule to within $$0.014$$ 0.014 Å of the full configuration interaction value. Within the same experiment, we estimate the polarizability of the H $${}_{2}$$ 2 molecule, finding agreement at the equilibrium bond length to within $$0.06$$ 0.06 a.u. ( $$2 \%$$ 2% relative error).

DOI
10.1038/s41534-019-0213-4

Related papers

Efficient quantum computation of molecular forces and other energy gradients

Thomas E. O’Brien, Michael Streif, Nicholas C. Rubin +12 more

While most work on the quantum simulation of chemistry has focused on computing energy surfaces, a similarly important application requiring subtly different algorithms is the computation of energy derivatives. Almost all molecular properties can be expressed an energy derivative, including molecular forces, which are essential for applications such as molecular dynamics simulations. Here, we introduce new quantum algorithms for computing molecular energy derivatives with significantly lower complexity than prior methods. Under cost models appropriate for noisy-intermediate scale quantum devices, we demonstrate how low-rank factorization and other tomography schemes can be optimized for energy derivative calculations. We numerically demonstrate that our techniques reduce the number of circuit repetitions required by many orders of magnitude for even modest systems, and that the cost of estimating an entire force vector may in some systems be lower than the cost of estimating the energy. In the context of fault-tolerant algorithms, we develop new methods of estimating energy derivatives with Heisenberg limited scaling, incorporating state-of-the-art techniques for block encoding fermionic operators. In contrast to our near-term results, we find that the cost of estimating forces with any of our Heisenberg-limited methods is bounded by the cost of estimating energies, due to inner loops requiring either energy estimation or reflections around the ground state. This implies that applications such as geometry optimization, coupling parameter estimation, and spectral prediction may be practical on fault-tolerant quantum devices, but tractable molecular dynamics simulations of large-scale systems requires further algorithmic advances.

arXiv:2111.12437•Physical Review Research•Dec 26, 2022•53 citations

Geometric quantum adiabatic methods for quantum chemistry

Hongye Yu, Deyu Lu, Qin Wu +1 more

Quantum algorithms have been successfully applied in quantum chemistry to obtain the ground-state energy of small molecules. Although accurate near the equilibrium geometry, the results can become unreliable when the chemical bonds are broken at large distances. For any adiabatic approach, this is usually caused by serious issues, such as energy gap closing or level crossing along the adiabatic evolution path. In this work, we propose a quantum algorithm based on adiabatic evolution to obtain molecular eigenstates and eigenenergies in quantum chemistry, which exploits a smooth geometric deformation by continuously varying bond lengths and bond angles. We demonstrate its utility in several examples on a noiseless quantum simulator, including ${\mathrm{H}}_{2}\mathrm{O}, {\mathrm{CH}}_{2}$, and a chemical reaction of ${\mathrm{H}}_{2}+{\mathrm{D}}_{2}\ensuremath{\rightarrow}$ 2HD, by uniformly stretching chemical bonds. We find that this new algorithm solves the problems related to energy gap closing and level crossing along the adiabatic evolution path at large atomic distances. The new method performs more stably and achieves better accuracy than our previous adiabatic method [Yu and Wei, Phys. Rev. Research 3, 013104 (2021)]. Furthermore, our fidelity analysis demonstrates that even with finite bond length changes, our algorithm still achieves high fidelity for the ground state.

arXiv:2112.15186•Physical Review Research•Jul 15, 2022•4 citations

Analytical Formulation of the Second-Order Derivative of Energy for the Orbital-Optimized Variational Quantum Eigensolver: Application to Polarizability

Yuya O. Nakagawa, Jiabao Chen, Shotaro Sudo +2 more

We develop a quantum-classical hybrid algorithm to calculate the analytical second-order derivative of the energy for the orbital-optimized variational quantum eigensolver (OO-VQE), which is a method to calculate eigenenergies of a given molecular Hamiltonian by utilizing near-term quantum computers and classical computers. We show that all quantities required in the algorithm to calculate the derivative can be evaluated on quantum computers as standard quantum expectation values without using any ancillary qubits. We validate our formula by numerical simulations of quantum circuits for computing the polarizability of the water molecule, which is the second-order derivative of the energy, with respect to the electric field. Moreover, the polarizabilities and refractive indices of thiophene and furan molecules are calculated as a test bed for possible industrial applications. We finally analyze the error scaling of the estimated polarizabilities obtained by the proposed analytical derivative versus the numerical derivative obtained by the finite difference. Numerical calculations suggest that our analytical derivative requires fewer measurements (runs) on quantum computers than the numerical derivative to achieve the same fixed accuracy.

arXiv:2211.03343•Journal of Chemical Theory and Computation•Mar 28, 2023•8 citations

Accurate and efficient calculations of Hellmann–Feynman forces for quantum computation

Juntao Lai, Yi Fan, Qiang Fu +2 more

First-order derivatives of energies with respect to atomic coordinates are widely computed and used in quantum chemistry simulations. The rapidly emerging technology of quantum computing offers a new paradigm for solving relevant quantum chemistry equations. In this work, we have achieved analytical calculations of atomic forces based on the Hellmann-Feynman theorem within the framework of the variational quantum eigensolver. The accuracy of the approach is demonstrated by calculating the atomic forces of H2, LiH, H2O, and NH3 molecules, which are in excellent agreement with values obtained from full configuration interaction calculations. In particular, for systems with degenerate molecular orbitals, the analytical approach has a significant accuracy advantage over finite-difference-based methods and will not involve additional computational effort on a quantum computer. The calculated forces are further used to optimize the geometries of NH3 and CH4 molecules and to perform ab initio molecular dynamics simulations for the umbrella inversion of NH3, demonstrating the feasibility of the approach in practical quantum chemistry simulations.

The Journal of Chemical Physics•Sep 19, 2023•14 citations

Quantum computing applied to calculations of molecular energies: CH2 benchmark

Libor Veis, Jiří Pittner

Quantum computers are appealing for their ability to solve some tasks much faster than their classical counterparts. It was shown in [Aspuru-Guzik et al., Science 309, 1704 (2005)] that they, if available, would be able to perform the full configuration interaction (FCI) energy calculations with a polynomial scaling. This is in contrast to conventional computers where FCI scales exponentially. We have developed a code for simulation of quantum computers and implemented our version of the quantum FCI algorithm. We provide a detailed description of this algorithm and the results of the assessment of its performance on the four lowest lying electronic states of CH(2) molecule. This molecule was chosen as a benchmark, since its two lowest lying (1)A(1) states exhibit a multireference character at the equilibrium geometry. It has been shown that with a suitably chosen initial state of the quantum register, one is able to achieve the probability amplification regime of the iterative phase estimation algorithm even in this case.

arXiv:1008.3451•The Journal of Chemical Physics•Nov 17, 2010•66 citations

End-to-End Simulation of Chemical Dynamics on a Quantum Computer

Elliot C. Eklund, Arkin Tikku, Patrick Sinnott +4 more

Simulations of chemical dynamics are a powerful means for understanding chemistry. However, classical computers struggle to simulate many chemical processes, especially non-adiabatic ones, where the Born-Oppenheimer approximation breaks down. Quantum computers could simulate quantum-chemical dynamics more efficiently than classical computers, but there is currently no complete quantum algorithm for calculating dynamical observables to within a known error. Here, we develop an efficient, end-to-end quantum algorithm for simulating chemical dynamics that avoids all uncontrolled approximations (including the Born-Oppenheimer approximation) and whose error is bounded subject to mild assumptions. To do so, we treat the nuclei and the electrons on an equal footing and simulate the full molecular wavefunction on a momentum-space grid in first quantization, including all algorithmic steps: initial-state preparation, time evolution using qubitization, and measurement of chemical observables such as reaction yields and rates. Our work gives the first algorithm for quantum simulation of chemistry whose end-to-end complexity achieves sublinear scaling in the size of the grid. We achieve this by developing an exponentially faster method for initial-state-preparation. Photochemistry is a likely early application of our algorithm and we estimate resources required for end-to-end simulations of non-adiabatic dynamics of atmospherically important molecules. Classically intractable photochemical computations could be performed using resources comparable to those required for other chemical applications of quantum computing.

quant-pharXiv:2603.19007•Mar 20, 2026•0 citations
Abstract Modeling chemical reactions and complicated molecular systems has been proposed as the “killer application” of a future quantum computer. Accurate calculations of derivatives of molecular eigenenergies are essential toward this end, allowing for geometry optimization, transition state searches, predictions of the response to an applied electric or magnetic field, and molecular dynamics simulations. In this work, we survey methods to calculate energy derivatives, and present two new methods: one based on quantum phase estimation, the other on a low-order response approximation. We calculate asymptotic error bounds and approximate computational scalings for the methods presented. Implementing these methods, we perform geometry optimization on an experimental quantum processor, estimating the equilibrium bond length of the dihydrogen molecule to within 0.0140.0140.014 0.014 Å of the full configuration interaction value. Within the same experiment, we estimate the polarizability of the H 2{}_{2}2​ 2 molecule, finding agreement at the equilibrium bond length to within 0.060.060.06 0.06 a.u. ( 2%2 \%2% 2% relative error).
While most work on the quantum simulation of chemistry has focused on computing energy surfaces, a similarly important application requiring subtly different algorithms is the computation of energy derivatives. Almost all molecular properties can be expressed an energy derivative, including molecular forces, which are essential for applications such as molecular dynamics simulations. Here, we introduce new quantum algorithms for computing molecular energy derivatives with significantly lower complexity than prior methods. Under cost models appropriate for noisy-intermediate scale quantum devices, we demonstrate how low-rank factorization and other tomography schemes can be optimized for energy derivative calculations. We numerically demonstrate that our techniques reduce the number of circuit repetitions required by many orders of magnitude for even modest systems, and that the cost of estimating an entire force vector may in some systems be lower than the cost of estimating the energy. In the context of fault-tolerant algorithms, we develop new methods of estimating energy derivatives with Heisenberg limited scaling, incorporating state-of-the-art techniques for block encoding fermionic operators. In contrast to our near-term results, we find that the cost of estimating forces with any of our Heisenberg-limited methods is bounded by the cost of estimating energies, due to inner loops requiring either energy estimation or reflections around the ground state. This implies that applications such as geometry optimization, coupling parameter estimation, and spectral prediction may be practical on fault-tolerant quantum devices, but tractable molecular dynamics simulations of large-scale systems requires further algorithmic advances.
Quantum algorithms have been successfully applied in quantum chemistry to obtain the ground-state energy of small molecules. Although accurate near the equilibrium geometry, the results can become unreliable when the chemical bonds are broken at large distances. For any adiabatic approach, this is usually caused by serious issues, such as energy gap closing or level crossing along the adiabatic evolution path. In this work, we propose a quantum algorithm based on adiabatic evolution to obtain molecular eigenstates and eigenenergies in quantum chemistry, which exploits a smooth geometric deformation by continuously varying bond lengths and bond angles. We demonstrate its utility in several examples on a noiseless quantum simulator, including H2O,CH2{\mathrm{H}}_{2}\mathrm{O}, {\mathrm{CH}}_{2}H2​O,CH2​, and a chemical reaction of ${\mathrm{H}}_{2}+{\mathrm{D}}_{2}\ensuremath{\rightarrow}$ 2HD, by uniformly stretching chemical bonds. We find that this new algorithm solves the problems related to energy gap closing and level crossing along the adiabatic evolution path at large atomic distances. The new method performs more stably and achieves better accuracy than our previous adiabatic method [Yu and Wei, Phys. Rev. Research 3, 013104 (2021)]. Furthermore, our fidelity analysis demonstrates that even with finite bond length changes, our algorithm still achieves high fidelity for the ground state.
We develop a quantum-classical hybrid algorithm to calculate the analytical second-order derivative of the energy for the orbital-optimized variational quantum eigensolver (OO-VQE), which is a method to calculate eigenenergies of a given molecular Hamiltonian by utilizing near-term quantum computers and classical computers. We show that all quantities required in the algorithm to calculate the derivative can be evaluated on quantum computers as standard quantum expectation values without using any ancillary qubits. We validate our formula by numerical simulations of quantum circuits for computing the polarizability of the water molecule, which is the second-order derivative of the energy, with respect to the electric field. Moreover, the polarizabilities and refractive indices of thiophene and furan molecules are calculated as a test bed for possible industrial applications. We finally analyze the error scaling of the estimated polarizabilities obtained by the proposed analytical derivative versus the numerical derivative obtained by the finite difference. Numerical calculations suggest that our analytical derivative requires fewer measurements (runs) on quantum computers than the numerical derivative to achieve the same fixed accuracy.
First-order derivatives of energies with respect to atomic coordinates are widely computed and used in quantum chemistry simulations. The rapidly emerging technology of quantum computing offers a new paradigm for solving relevant quantum chemistry equations. In this work, we have achieved analytical calculations of atomic forces based on the Hellmann-Feynman theorem within the framework of the variational quantum eigensolver. The accuracy of the approach is demonstrated by calculating the atomic forces of H2, LiH, H2O, and NH3 molecules, which are in excellent agreement with values obtained from full configuration interaction calculations. In particular, for systems with degenerate molecular orbitals, the analytical approach has a significant accuracy advantage over finite-difference-based methods and will not involve additional computational effort on a quantum computer. The calculated forces are further used to optimize the geometries of NH3 and CH4 molecules and to perform ab initio molecular dynamics simulations for the umbrella inversion of NH3, demonstrating the feasibility of the approach in practical quantum chemistry simulations.
Quantum computers are appealing for their ability to solve some tasks much faster than their classical counterparts. It was shown in [Aspuru-Guzik et al., Science 309, 1704 (2005)] that they, if available, would be able to perform the full configuration interaction (FCI) energy calculations with a polynomial scaling. This is in contrast to conventional computers where FCI scales exponentially. We have developed a code for simulation of quantum computers and implemented our version of the quantum FCI algorithm. We provide a detailed description of this algorithm and the results of the assessment of its performance on the four lowest lying electronic states of CH(2) molecule. This molecule was chosen as a benchmark, since its two lowest lying (1)A(1) states exhibit a multireference character at the equilibrium geometry. It has been shown that with a suitably chosen initial state of the quantum register, one is able to achieve the probability amplification regime of the iterative phase estimation algorithm even in this case.
Simulations of chemical dynamics are a powerful means for understanding chemistry. However, classical computers struggle to simulate many chemical processes, especially non-adiabatic ones, where the Born-Oppenheimer approximation breaks down. Quantum computers could simulate quantum-chemical dynamics more efficiently than classical computers, but there is currently no complete quantum algorithm for calculating dynamical observables to within a known error. Here, we develop an efficient, end-to-end quantum algorithm for simulating chemical dynamics that avoids all uncontrolled approximations (including the Born-Oppenheimer approximation) and whose error is bounded subject to mild assumptions. To do so, we treat the nuclei and the electrons on an equal footing and simulate the full molecular wavefunction on a momentum-space grid in first quantization, including all algorithmic steps: initial-state preparation, time evolution using qubitization, and measurement of chemical observables such as reaction yields and rates. Our work gives the first algorithm for quantum simulation of chemistry whose end-to-end complexity achieves sublinear scaling in the size of the grid. We achieve this by developing an exponentially faster method for initial-state-preparation. Photochemistry is a likely early application of our algorithm and we estimate resources required for end-to-end simulations of non-adiabatic dynamics of atmospherically important molecules. Classically intractable photochemical computations could be performed using resources comparable to those required for other chemical applications of quantum computing.