Toward Accurate Post-Born-Oppenheimer Molecular Simulations on Quantum Computers: An Adaptive Variational Eigensolver with Nuclear-Electronic Frozen Natural Orbitals.

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 H2 and D2 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.

Nonadiabatic Nuclear-Electron Dynamics: A Quantum Computing Approach.

Coupled quantum electron-nuclear dynamics is often associated with the Born-Huang expansion of the molecular wave function and the appearance of nonadiabatic effects as a perturbation. On the other hand, native multicomponent representations of electrons and nuclei also exist, which do not rely on any a priori approximation. However, their implementation is hampered by prohibitive scaling. Consequently, quantum computers offer a unique opportunity for extending their use to larger systems. Here, we propose a quantum algorithm for simulating the time-evolution of molecular systems and apply it to proton transfer dynamics in malonaldehyde, described as a rigid scaffold. The proposed quantum algorithm can be easily generalized to include the explicit dynamics of the classically described molecular scaffold. We show how entanglement between electronic and nuclear degrees of freedom can persist over long times if electrons do not follow the nuclear displacement adiabatically. The proposed quantum algorithm may become a valid candidate for the study of such phenomena when sufficiently powerful quantum computers become available.

A quantum computing implementation of nuclearelectronic orbital (NEO) theory: Toward an exact pre-Born-Oppenheimer formulation of molecular quantum systems.

Nuclear quantum phenomena beyond the Born-Oppenheimer approximation are known to play an important role in a growing number of chemical and biological processes. While there exists no unique consensus on a rigorous and efficient implementation of coupled electron-nuclear quantum dynamics, it is recognized that these problems scale exponentially with system size on classical processors and, therefore, may benefit from quantum computing implementations. Here, we introduce a methodology for the efficient quantum treatment of the electron-nuclear problem on near-term quantum computers, based upon the Nuclear-Electronic Orbital (NEO) approach. We generalize the electronic two-qubit tapering scheme to include nuclei by exploiting symmetries inherent in the NEO framework, thereby reducing the Hamiltonian dimension, number of qubits, gates, and measurements needed for calculations. We also develop parameter transfer and initialization techniques, which improve convergence behavior relative to conventional initialization. These techniques are applied to H2 and malonaldehyde for which results agree with NEO full configuration interaction and NEO complete active space configuration interaction benchmarks for ground state energy to within 10-6 hartree and entanglement entropy to within 10-4. These implementations therefore significantly reduce resource requirements for full quantum simulations of molecules on near-term quantum devices while maintaining high accuracy.

Self-adaptive tensor network states with multi-site correlators.

We introduce the concept of self-adaptive tensor network states (SATNSs) based on multi-site correlators. The SATNS ansatz gradually extends its variational space incorporating the most important next-order correlators into the ansatz for the wave function. The selection of these correlators is guided by entanglement-entropy measures from quantum information theory. By sequentially introducing variational parameters and adjusting them to the system under study, the SATNS ansatz achieves keeping their number significantly smaller than the total number of full-configuration interaction parameters. The SATNS ansatz is studied for manganocene in its lowest-energy sextet and doublet states; the latter of which is known to be difficult to describe. It is shown that the SATNS parametrization solves the convergence issues found for previous correlator-based tensor network states.

New Approaches for ab initio Calculations of Molecules with Strong Electron Correlation.

Reliable quantum chemical methods for the description of molecules with dense-lying frontier orbitals are needed in the context of many chemical compounds and reactions. Here, we review developments that led to our new computational toolbox which implements the quantum chemical density matrix renormalization group in a second-generation algorithm. We present an overview of the different components of this toolbox.

Analytical gradients for excitation energies from frozen-density embedding.

The formulation of analytical excitation-energy gradients from time-dependent density functional theory within the frozen-density embedding framework is presented. In addition to a comprehensive mathematical derivation, we discuss details of the numerical implementation in the Slater-function based Amsterdam Density Functional (ADF) program. Particular emphasis is put on the consistency in the use of approximations for the evaluation of second- and third-order non-additive kinetic-energy and exchange-correlation functional derivatives appearing in the final expression for the excitation-energy gradient. We test the implementation for different chemical systems in which molecular excited-state potential-energy curves are affected by another subsystem. It is demonstrated that the analytical implementation for the evaluation of excitation-energy gradients yields results in close agreement with data from numerical differentiation. In addition, we show that our analytical results are numerically more stable and thus preferable over the numerical ones.

Selective TDDFT with automatic removal of ghost transitions: application to a perylene-dye-sensitized solar cell model.

We present an application of a selective time-dependent density-functional theory (TDDFT) scheme to a model for a dye-sensitized solar cell (DSSC) with a perylene sensitizer dye on a TiO2 nanoparticle model. In an earlier study on this system [De Angelis, Chem. Phys. Lett., 2010, 493, 323], it was reported that a large number of conduction-band excitations severely complicate the identification of the bright π→π* excitations of the perylene dye. Here, we show that this problem can be overcome by applying a selective TDDFT solver based on a guess for the relevant orbital transition in combination with a suitable root-homing scheme. In order to enhance the efficiency of this algorithm we implement an automatic removal scheme for artificially low-lying long-range charge-transfer transitions from the TDDFT eigenvalue problem. A large number of such transitions appear in explicitly solvated systems in the form of inter-solvent or solvent-solute transitions. We study the characteristics of this removal scheme for a small water cluster and then apply it in a TDDFT calculation to a perylene-TiO2 nanoparticle model system and to perylene explicitly solvated in methanol. It is demonstrated that this scheme leads to a large reduction in the computational cost with essentially no loss in accuracy. Large differences in the effect of adsorption on the excited states of perylene dyes with two different anchor groups found in earlier work are confirmed.

State-selective optimization of local excited electronic states in extended systems.

Standard implementations of time-dependent density-functional theory (TDDFT) for the calculation of excitation energies give access to a number of the lowest-lying electronic excitations of a molecule under study. For extended systems, this can become cumbersome if a particular excited state is sought-after because many electronic transitions may be present. This often means that even for systems of moderate size, a multitude of excited states needs to be calculated to cover a certain energy range. Here, we present an algorithm for the selective determination of predefined excited electronic states in an extended system. A guess transition density in terms of orbital transitions has to be provided for the excitation that shall be optimized. The approach employs root-homing techniques together with iterative subspace diagonalization methods to optimize the electronic transition. We illustrate the advantages of this method for solvated molecules, core-excitations of metal complexes, and adsorbates at cluster surfaces. In particular, we study the local π→π(∗) excitation of a pyridine molecule adsorbed at a silver cluster. It is shown that the method works very efficiently even for high-lying excited states. We demonstrate that the assumption of a single, well-defined local excitation is, in general, not justified for extended systems, which can lead to root-switching during optimization. In those cases, the method can give important information about the spectral distribution of the orbital transition employed as a guess.