We introduce a novel approach for a fully quantum description of coupled electron–ion systems from first principles. It combines the variational quantum Monte Carlo solution of the electronic part with the path integral formalism for the quantum nuclear dynamics. On the one hand, the path integral molecular dynamics includes nuclear quantum effects by adding a set of fictitious classical particles (beads) aimed at reproducing nuclear quantum fluctuations via a harmonic kinetic term. On the other hand, variational quantum Monte Carlo can provide Born–Oppenheimer potential energy surfaces with a precision comparable to the most-advanced post-Hartree–Fock approaches, and with a favorable scaling with the system size. In order to cope with the intrinsic noise due to the stochastic nature of quantum Monte Carlo methods, we generalize the path integral molecular dynamics using a Langevin thermostat correlated according to the covariance matrix of quantum Monte Carlo nuclear forces. The variational parameters of the quantum Monte Carlo wave function are evolved during the nuclear dynamics, such that the Born–Oppenheimer potential energy surface is unbiased. Statistical errors on the wave function parameters are reduced by resorting to bead grouping average, which we show to be accurate and well-controlled. Our general algorithm relies on a Trotter breakup between the dynamics driven by ionic forces and the one set by the harmonic interbead couplings. The latter is exactly integrated, even in the presence of the Langevin thermostat, thanks to the mapping onto an Ornstein–Uhlenbeck process. This framework turns out to be also very efficient in the case of noiseless (deterministic) ionic forces. The new implementation is validated on the Zundel ion (H 5 O 2 + ) by direct comparison with standard path integral Langevin dynamics calculations made with a coupled cluster potential energy surface. Nuclear quantum effects are confirmed to be dominant over thermal effects well beyond room temperature, giving the excess proton an increased mobility by quantum tunneling.
We introduce a novel approach for a fully quantum description of coupled electron–ion systems from first principles. It combines the variational quantum Monte Carlo solution of the electronic part with the path integral formalism for the quantum nuclear dynamics. On the one hand, the path integral molecular dynamics includes nuclear quantum effects by adding a set of fictitious classical particles (beads) aimed at reproducing nuclear quantum fluctuations via a harmonic kinetic term. On the other hand, variational quantum Monte Carlo can provide Born–Oppenheimer potential energy surfaces with a precision comparable to the most-advanced post-Hartree–Fock approaches, and with a favorable scaling with the system size. In order to cope with the intrinsic noise due to the stochastic nature of quantum Monte Carlo methods, we generalize the path integral molecular dynamics using a Langevin thermostat correlated according to the covariance matrix of quantum Monte Carlo nuclear forces. The variational parameters of the quantum Monte Carlo wave function are evolved during the nuclear dynamics, such that the Born–Oppenheimer potential energy surface is unbiased. Statistical errors on the wave function parameters are reduced by resorting to bead grouping average, which we show to be accurate and well-controlled. Our general algorithm relies on a Trotter breakup between the dynamics driven by ionic forces and the one set by the harmonic interbead couplings. The latter is exactly integrated, even in the presence of the Langevin thermostat, thanks to the mapping onto an Ornstein–Uhlenbeck process. This framework turns out to be also very efficient in the case of noiseless (deterministic) ionic forces. The new implementation is validated on the Zundel ion (H 5 O 2 + ) by direct comparison with standard path integral Langevin dynamics calculations made with a coupled cluster potential energy surface. Nuclear quantum effects are confirmed to be dominant over thermal effects well beyond room temperature, giving the excess proton an increased mobility by quantum tunneling.
ConspectusSimulating molecular dynamics (MD) within a comprehensive quantum framework has been a long-standing challenge in computational chemistry. An exponential scaling of computational cost renders solving the time dependent Schrödinger equation (TDSE) of a molecular Hamiltonian, including both electronic and nuclear degrees of freedom (DOFs), as well as their couplings, infeasible for more than a few DOFs. In the Born-Oppenheimer (BO), or adiabatic, picture, electronic and nuclear parts of the wave function are decoupled and treated separately. Within this framework, the nuclear wave function evolves along potential energy surfaces (PESs) computed as solutions to the electronic Schrödinger equation parametrized in the nuclear DOFs. This approximation, together with increasingly elaborate numerical approaches to solve the nuclear time dependent Schrödinger equation (TDSE), enabled the treatment of up to a few dozens of degrees of freedom (DOFs). However, for particular applications, such as photochemistry, the BO approximation breaks down. In this regime of non-adiabatic dynamics, solving the full molecular problem including electron-nuclear couplings becomes essential, further increasing the complexity of the numerical solution. Although valuable methods such as multiconfigurational time-dependent Hartree (MCTDH) have been proposed for the solution of the coupled electron-nuclear dynamics, they remain hampered by an exponential scaling in the number of nuclear DOFs and by the difficulty of finding universal variational forms.In this Account, we present a perspective on novel quantum computational algorithms, aiming to alleviate the exponential scaling inherent to the simulation of many-body quantum dynamics. In particular, we focus on the derivation and application of quantum algorithms for adiabatic and non-adiabatic quantum dynamics, which include efficient approaches for the calculation of the BO potential energy surfaces (PESs). Thereafter, we study the time-evolution of a model system consisting of two coupled PESs in first and second quantization. In a first application, we discuss a recently introduced quantum algorithm for the evolution of a wavepacket in first quantization and exploit the potential quantum advantage of mapping its spatial grid representation to logarithmically many qubits. For the second demonstration, we move to the second quantization framework and review the scaling properties of two alternative time-evolution algorithms, namely, a variational quantum algorithm (VQA) (based on the McLachlan variational principle) and conventional Trotter-type evolution (based on a Lie-Trotter-Suzuki formula). Both methods clearly demonstrate the potential of quantum algorithms and their favorable scaling compared to the available classical approaches. However, a clear demonstration of quantum advantage in the context of molecular quantum dynamics may require the implementation of these algorithms in fault-tolerant quantum computers, while their application in near-term, noisy quantum devices is still unclear and deserves further investigation.
Nuclear quantum effects are critical for describing proton transfer and hydrogen bonding, but their incorporation into quantum chemistry calculations is often computationally prohibitive on classical hardware. A promising alternative are quantum computers due to their linear scaling in the space requirements with system size. We introduce a novel hybrid quantum-classical algorithm for ab-initio molecular dynamics that incorporates nuclear quantum effects via the nuclear-electronic orbital method. The proposed algorithm evaluates ground state energies and forces on the quantum computer using a variational quantum eigensolver, while the molecular geometries are updated classically. We validate our approach through simulations of H2, H2O and the Zundel ion H5O2+, comparing the simulated vibrational spectra with experimental data. Upon inclusion of nuclear quantum effects, the proton shuttling movement in the Zundel ion becomes effectively barrierless, and errors in the simulated frequencies improve significantly. Employing compact hardware-efficient ansatz circuits we achieve results comparable to the more accurate UCCSD ansatzes, which hints towards the feasibility of executing our algorithm on near-term quantum devices.
Nuclear quantum effects such as zero-point energy and hydrogen tunneling play a central role in many biological and chemical processes. The nuclear-electronic orbital (NEO) approach captures these effects by treating selected nuclei quantum mechanically on the same footing as electrons. On classical computers, the resources required for an exact solution of NEO-based models grow exponentially with system size. By contrast, quantum computers offer a means of solving this problem with polynomial scaling. However, due to the limitations of current quantum devices, NEO simulations are confined to the smallest systems described by minimal basis sets, whereas realistic simulations beyond the Born–Oppenheimer approximation require more sophisticated basis sets. For this purpose, we herein extend a hardware-efficient ADAPT-VQE method to the NEO framework in the frozen natural orbital (FNO) basis. We demonstrate on H 2 and D 2 molecules that the NEO-FNO-ADAPT-VQE method reduces the CNOT count by several orders of magnitude relative to the NEO unitary coupled cluster method with singles and doubles while maintaining the desired accuracy. This extreme reduction in the CNOT gate count is sufficient to permit practical computations employing the NEO method─an important step toward accurate simulations involving nonclassical nuclei and non-Born–Oppenheimer effects on near-term quantum devices. We further show that the method can capture isotope effects, and we demonstrate that inclusion of correlation energy systematically improves the prediction of difference in the zero-point energy (ΔZPE) between isotopes.
We study the efficiency, precision and accuracy of all-electron variational and diffusion quantum Monte Carlo calculations using Slater basis sets. Starting from wave functions generated by Hartree-Fock and density functional theory, we describe an algorithm to enforce the electron-nucleus cusp condition by linear projection. For the 55 molecules in the G2 set, the diffusion quantum Monte Carlo calculations recovers an average of 95% of the correlation energy and reproduces bond energies to a mean absolute deviation of 3.2 kcal/mol. Comparing the individual total energies with essentially exact values, we investigate the error cancellation in atomization and chemical reaction path energies, giving additional insight into the sizes of nodal surface errors.
Quantum Monte Carlo methods are accurate and promising many body techniques for electronic structure calculations which, in the last years, are encountering a growing interest thanks to their favorable scaling with the system size and their efficient parallelization, particularly suited for the modern high performance computing facilities. The ansatz of the wave function and its variational flexibility are crucial points for both the accurate description of molecular properties and the capabilities of the method to tackle large systems. In this paper, we extensively analyze, using different variational ansatzes, several properties of the water molecule, namely, the total energy, the dipole and quadrupole momenta, the ionization and atomization energies, the equilibrium configuration, and the harmonic and fundamental frequencies of vibration. The investigation mainly focuses on variational Monte Carlo calculations, although several lattice regularized diffusion Monte Carlo calculations are also reported. Through a systematic study, we provide a useful guide to the choice of the wave function, the pseudopotential, and the basis set for QMC calculations. We also introduce a new method for the computation of forces with finite variance on open systems and a new strategy for the definition of the atomic orbitals involved in the Jastrow-Antisymmetrised Geminal power wave function, in order to drastically reduce the number of variational parameters. This scheme significantly improves the efficiency of QMC energy minimization in case of large basis sets.
Quantum trajectory calculations for electrons are a useful tool in the field of molecular dynamics, e.g. to understand processes in ultrafast spectroscopy. They have, however, two limitation: On the one hand, such calculations are typically based on the Born-Oppenheimer approximation (BOA) and the electron dynamics for stationary nuclei is considered, thus neglecting quantum effects of the nuclei. On the other hand, even if the quantum nuclear motion would be taken into account, a BOA dynamics on a single potential energy surface would not provide any electron trajectories because the electronic part is treated as a stationary problem. By using the exact factorization method, we overcome these limitations and generalize the theory of electronic quantum trajectories to a fully quantum-mechanical treatment of the nuclei. After reviewing the time-dependent theory of quantum hydrodynamics and quantum trajectories, we show that the nuclei can be viewed as a quantum clock for the electronic motion and we develop a fully quantum-mechanical clock-dependent version of quantum hydrodynamics. This theory is used to obtain electronic trajectories for quantum nuclei, as is exemplified for a model system of a proton-coupled electron transfer dynamics. Our work generalizes the concept of quantum trajectories and lays the foundations for the development of trajectory-based simulation methods of electron dynamics beyond the BOA.