Reducing the Runtime of Fault-Tolerant Quantum Simulations in Chemistry through Symmetry-Compressed Double Factorization.

Quantum phase estimation based on qubitization is the state-of-the-art fault-tolerant quantum algorithm for computing ground-state energies in chemical applications. In this context, the 1-norm of the Hamiltonian plays a fundamental role in determining the total number of required iterations and also the overall computational cost. In this work, we introduce the symmetry-compressed double factorization (SCDF) approach, which combines a CDF of the Hamiltonian with the symmetry shift technique, significantly reducing the 1-norm value. The effectiveness of this approach is demonstrated numerically by considering various benchmark systems, including the FeMoco molecule, cytochrome P450, and hydrogen chains of different sizes. To compare the efficiency of SCDF to other methods in absolute terms, we estimate Toffoli gate requirements, which dominate the execution time on fault-tolerant quantum computers. For the systems considered here, SCDF leads to a sizable reduction of the Toffoli gate count in comparison to other variants of DF or even tensor hypercontraction, which is usually regarded as the most efficient approach for qubitization.

Estimation of Electrostatic Interaction Energies on a Trapped-Ion Quantum Computer.

We present the first hardware implementation of electrostatic interaction energies by using a trapped-ion quantum computer. As test system for our computation, we focus on the reduction of NO to N2O catalyzed by a nitric oxide reductase (NOR). The quantum computer is used to generate an approximate ground state within the NOR active space. To efficiently measure the necessary one-particle density matrices, we incorporate fermionic basis rotations into the quantum circuit without extending the circuit length, laying the groundwork for further efficient measurement routines using factorizations. Measurements in the computational basis are then used as inputs for computing the electrostatic interaction energies on a classical computer. Our experimental results strongly agree with classical noise-less simulations of the same circuits, finding electrostatic interaction energies within chemical accuracy despite hardware noise. This work shows that algorithms tailored to specific observables of interest, such as interaction energies, may require significantly fewer quantum resources than individual ground state energies would require in the straightforward supermolecular approach.

Accurate non-covalent interaction energies on noisy intermediate-scale quantum computers via second-order symmetry-adapted perturbation theory.

The calculation of non-covalent interaction energies on noisy intermediate-scale quantum (NISQ) computers appears to be challenging with straightforward application of existing quantum algorithms. For example, the use of the standard supermolecular method with the variational quantum eigensolver (VQE) would require extremely precise resolution of the total energies of the fragments to provide for accurate subtraction to the interaction energy. Here we present a symmetry-adapted perturbation theory (SAPT) method that may provide interaction energies with high quantum resource efficiency. Of particular note, we present a quantum extended random-phase approximation (ERPA) treatment of the SAPT second-order induction and dispersion terms, including exchange counterparts. Together with previous work on first-order terms (Chem. Sci., 2022, 13, 3094), this provides a recipe for complete SAPT(VQE) interaction energies up to second order, which is a well established truncation. The SAPT interaction energy terms are computed as first-level observables with no subtraction of monomer energies invoked, and the only quantum observations needed are the VQE one- and two-particle density matrices. We find empirically that SAPT(VQE) can provide accurate interaction energies even with coarsely optimized, low circuit depth wavefunctions from a quantum computer, simulated through ideal statevectors. The errors of the total interaction energy are orders of magnitude lower than the corresponding VQE total energy errors of the monomer wavefunctions. In addition, we present heme-nitrosyl model complexes as a system class for near term quantum computing simulations. They are strongly correlated, biologically relevant and difficult to simulate with classical quantum chemical methods. This is illustrated with density functional theory (DFT) as the predicted interaction energies exhibit a strong sensitivity with respect to the choice of functional. Thus, this work paves the way to obtain accurate interaction energies on a NISQ-era quantum computer with few quantum resources. It is the first step in alleviating one of the major challenges in quantum chemistry, where in-depth knowledge of both the method and system is required a priori to reliably generate accurate interaction energies.

Efficient quantum analytic nuclear gradients with double factorization.

Efficient representations of the Hamiltonian, such as double factorization, drastically reduce the circuit depth or the number of repetitions in error corrected and noisy intermediate-scale quantum (NISQ) algorithms for chemistry. We report a Lagrangian-based approach for evaluating relaxed one- and two-particle reduced density matrices from double factorized Hamiltonians, unlocking efficiency improvements in computing the nuclear gradient and related derivative properties. We demonstrate the accuracy and feasibility of our Lagrangian-based approach to recover all off-diagonal density matrix elements in classically simulated examples with up to 327 quantum and 18 470 total atoms in QM/MM simulations with modest-sized quantum active spaces. We show this in the context of the variational quantum eigensolver in case studies, such as transition state optimization, ab initio molecular dynamics simulation, and energy minimization of large molecular systems.

Towards the simulation of large scale protein-ligand interactions on NISQ-era quantum computers.

We explore the use of symmetry-adapted perturbation theory (SAPT) as a simple and efficient means to compute interaction energies between large molecular systems with a hybrid method combining NISQ-era quantum and classical computers. From the one- and two-particle reduced density matrices of the monomer wavefunctions obtained by the variational quantum eigensolver (VQE), we compute SAPT contributions to the interaction energy [SAPT(VQE)]. At first order, this energy yields the electrostatic and exchange contributions for non-covalently bound systems. We empirically find from ideal statevector simulations that the SAPT(VQE) interaction energy components display orders of magnitude lower absolute errors than the corresponding VQE total energies. Therefore, even with coarsely optimized low-depth VQE wavefunctions, we still obtain sub kcal mol-1 accuracy in the SAPT interaction energies. In SAPT(VQE), the quantum requirements, such as qubit count and circuit depth, are lowered by performing computations on the separate molecular systems. Furthermore, active spaces allow for large systems containing thousands of orbitals to be reduced to a small enough orbital set to perform the quantum portions of the computations. We benchmark SAPT(VQE) (with the VQE component simulated by ideal statevector simulators) against a handful of small multi-reference dimer systems and the iron center containing human cancer-relevant protein lysine-specific demethylase 5 (KDM5A).

Rank-reduced coupled-cluster. III. Tensor hypercontraction of the doubles amplitudes.

We develop a quartic-scaling implementation of coupled-cluster singles and doubles (CCSD) based on low-rank tensor hypercontraction (THC) factorizations of both the electron repulsion integrals (ERIs) and the doubles amplitudes. This extends our rank-reduced (RR) coupled-cluster method to incorporate higher-order tensor factorizations. The THC factorization of the doubles amplitudes accounts for most of the gain in computational efficiency as it is sufficient, in conjunction with a Cholesky decomposition of the ERIs, to reduce the computational complexity of most contributions to the CCSD amplitude equations. Further THC factorization of the ERIs reduces the complexity of certain terms arising from nested commutators between the doubles excitation operator and the two-electron operator. We implement this new algorithm using graphical processing units and demonstrate that it enables CCSD calculations for molecules with 250 atoms and 2500 basis functions using a single computer node. Furthermore, we show that the new method computes correlation energies with comparable accuracy to the underlying RR-CCSD method.

An ab initio exciton model for singlet fission.

We present an ab initio exciton model that extends the Frenkel exciton model and includes valence, charge-transfer, and multiexcitonic excited states. It serves as a general, parameter-free, yet computationally efficient and scalable approach for simulation of singlet fission processes in multichromophoric systems. A comparison with multiconfigurational methods confirms that our exciton model predicts consistent energies and couplings for the pentacene dimer and captures the correct physics. Calculations of larger pentacene clusters demonstrate the computational scalability of the exciton model and suggest that the mixing between local and charge-transfer excitations narrows the gap between singlet and multiexcitonic states. Local vibrations of pentacene molecules are found to facilitate singlet-multiexcitonic state-crossing and hence are important for understanding singlet fission. The exciton model developed in this work also sets the stage for further implementation of the nuclear gradients and nonadiabatic couplings needed for first principles nonadiabatic quantum molecular dynamics simulations of singlet fission.

Nonadiabatic Dynamics of Photoexcited cis-Stilbene Using Ab Initio Multiple Spawning.

The photochemistry of cis-stilbene proceeds through two pathways: cis-trans isomerization and ring closure to 4a,4b-dihydrophenanthrene (DHP). Despite serving for many decades as a model system for photoisomerization, the photodynamics of cis-stilbene is still not fully understood. We use ab initio multiple spawning on a SA-2-CASSCF(2,2) potential energy surface to simulate the nonadiabatic dynamics of isolated cis-stilbene. We find the cyclization (to DHP and cis-stilbene) and isomerization (to trans- and cis-stilbene) reaction coordinates to be orthogonal; branching between the two pathways is determined on the S1 excited state within 150 fs of photoexcitation. Trajectory basis functions (TBFs) undergoing cyclization decay rapidly to the ground state in 250 fs, while TBFs moving along the isomerization coordinate remain on the excited state longer, with the majority decaying between 300 and 500 fs. We observe three avoided crossing regions in the dynamics: two along the isomerization coordinate (displaying pyramidalization and migration of an ethylenic hydrogen or phenyl group), and one DHP-like conical intersection along the cyclization coordinate. The isomeric form of the vibrationally hot photoproducts (as determined by measurement 2 ps after photoexcitation) is determined within less than 50 fs of decay to the ground state mediated by passage through a conical intersection. Excess vibrational energy of ground state cis- and trans-stilbene is channelled into phenyl torsions (with mostly opposing directionality). Our simulations are validated by direct comparison to experiment for the absorption spectrum, branching ratio of the three photoproducts (44:52:4 cis-stilbene:trans-stilbene:DHP), and excited state lifetime (520 ± 40 fs).

Psi4 1.4: Open-source software for high-throughput quantum chemistry.

PSI4 is a free and open-source ab initio electronic structure program providing implementations of Hartree-Fock, density functional theory, many-body perturbation theory, configuration interaction, density cumulant theory, symmetry-adapted perturbation theory, and coupled-cluster theory. Most of the methods are quite efficient, thanks to density fitting and multi-core parallelism. The program is a hybrid of C++ and Python, and calculations may be run with very simple text files or using the Python API, facilitating post-processing and complex workflows; method developers also have access to most of PSI4's core functionalities via Python. Job specification may be passed using The Molecular Sciences Software Institute (MolSSI) QCSCHEMA data format, facilitating interoperability. A rewrite of our top-level computation driver, and concomitant adoption of the MolSSI QCARCHIVE INFRASTRUCTURE project, makes the latest version of PSI4 well suited to distributed computation of large numbers of independent tasks. The project has fostered the development of independent software components that may be reused in other quantum chemistry programs.

Simultaneous observation of nuclear and electronic dynamics by ultrafast electron diffraction.

Simultaneous observation of nuclear and electronic motion is crucial for a complete understanding of molecular dynamics in excited electronic states. It is challenging for a single experiment to independently follow both electronic and nuclear dynamics at the same time. Here we show that ultrafast electron diffraction can be used to simultaneously record both electronic and nuclear dynamics in isolated pyridine molecules, naturally disentangling the two components. Electronic state changes (S1→S0 internal conversion) were reflected by a strong transient signal in small-angle inelastic scattering, and nuclear structural changes (ring puckering) were monitored by large-angle elastic diffraction. Supported by ab initio nonadiabatic molecular dynamics and diffraction simulations, our experiment provides a clear view of the interplay between electronic and nuclear dynamics of the photoexcited pyridine molecule.

Rank reduced coupled cluster theory. II. Equation-of-motion coupled-cluster singles and doubles.

Equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) is a reliable and popular approach to the determination of electronic excitation energies. Recently, we have developed a rank-reduced CCSD (RR-CCSD) method that allows the ground-state coupled-cluster energy to be determined with low-rank cluster amplitudes. Here, we extend this approach to excited-state energies through a RR-EOM-CCSD method. We start from the EOM-CCSD energy functional and insert low-rank approximations to the doubles amplitudes. The result is an approximate EOM-CCSD method with only a quadratic number (in the molecular size) of free parameters in the wavefunction. Importantly, our formulation of RR-EOM-CCSD preserves the size intensivity of the excitation energy and size extensivity of the total energy. Numerical tests of the method suggest that accuracy on the order of 0.05-0.01 eV in the excitation energy is possible with 1% or less of the original number of wavefunction coefficients; accuracy of better than 0.01 eV can be achieved with about 4% or less of the free parameters. The amount of compression at a given accuracy level is expected to increase with the size of the molecule. The RR-EOM-CCSD method is a new path toward the efficient determination of accurate electronic excitation energies.

Observation of Ultrafast Intersystem Crossing in Thymine by Extreme Ultraviolet Time-Resolved Photoelectron Spectroscopy.

We studied the photoinduced ultrafast relaxation dynamics of the nucleobase thymine using gas-phase time-resolved photoelectron spectroscopy. By employing extreme ultraviolet pulses from high harmonic generation for photoionization, we substantially extend our spectral observation window with respect to previous studies. This enables us to follow relaxation of the excited state population all the way to low-lying electronic states including the ground state. In thymine, we observe relaxation from the optically bright 1ππ* state of thymine to a dark 1nπ* state within 80 ± 30 fs. The 1nπ* state relaxes further within 3.5 ± 0.3 ps to a low-lying electronic state. By comparison with quantum chemical simulations, we can unambiguously assign its spectroscopic signature to the 3ππ* state. Hence, our study draws a comprehensive picture of the relaxation mechanism of thymine including ultrafast intersystem crossing to the triplet manifold.

Quantum Computation of Electronic Transitions Using a Variational Quantum Eigensolver.

We develop an extension of the variational quantum eigensolver (VQE) algorithm-multistate contracted VQE (MC-VQE)-that allows for the efficient computation of the transition energies between the ground state and several low-lying excited states of a molecule, as well as the oscillator strengths associated with these transitions. We numerically simulate MC-VQE by computing the absorption spectrum of an ab initio exciton model of an 18-chromophore light-harvesting complex from purple photosynthetic bacteria.

Perturbation of Short Hydrogen Bonds in Photoactive Yellow Protein via Noncanonical Amino Acid Incorporation.

Photoactive yellow protein (PYP) is a small photoreceptor protein that has two unusually short hydrogen bonds between the deprotonated p-coumaric acid chromophore and two amino acids, a tyrosine and a glutamic acid. This has led to considerable debate as to whether the glutamic acid-chromophore hydrogen bond is a low barrier hydrogen bond, with conflicting results in the literature. We have modified the p Ka of the tyrosine by amber suppression and of the chromophore by chemical substitution. X-ray crystal structures of these modified proteins are nearly identical to the wild-type protein, so the heavy atom distance between proton donor and acceptor is maintained, even though these modifications change the relative proton affinity between donor and acceptor. Despite a considerable change in relative proton affinity, the NMR chemical shifts of the hydrogen-bonded protons are only moderately affected. QM/MM calculations were used to explore the protons' potential energy surface and connect the calculated proton position with empirically measured proton chemical shifts. The results are inconsistent with a low barrier hydrogen bond but in all cases are consistent with a localized proton, suggesting an ionic hydrogen bond rather than a low barrier hydrogen bond.

Rank reduced coupled cluster theory. I. Ground state energies and wavefunctions.

We propose a compression of the opposite-spin coupled cluster doubles amplitudes of the form τij ab≡Uia VTVWUjb W, where Uia V are the nV-highest magnitude eigenvectors of the MP2 or MP3 doubles amplitudes. Together with a corresponding parameterization of the opposite-spin coupled cluster Lagrange multipliers of the form λab ij≡Uia VLVWUjb W, this yields a fully self-consistent parameterization of reduced-rank coupled cluster equations in terms of the Lagrangian L0TVW,LVW. Making this Lagrangian stationary with respect to the LVW parameters yields a perfectly determined set of equations for the TVW equations and coupled cluster energy. These equations can be solved using a Lyapunov equation for the first-order amplitude updates. We test this "rank-reduced coupled cluster" method for coupled cluster singles and doubles in medium sized molecules and find that substantial compression of the T^2 amplitudes is possible with acceptable accuracy.

Ab Initio Computation of Rotationally-Averaged Pump-Probe X-ray and Electron Diffraction Signals.

We develop a new algorithm for the computation of the rotationally averaged elastic molecular diffraction signal for the cases of perpendicular or parallel pump-probe geometries. The algorithm first collocates the charge density from an arbitrary ab initio wave function onto a Becke quadrature grid [A. Becke, J. Chem. Phys. 1988 , 88 , 2457 ], providing a high-fidelity multiresolution representation of the charge density. A double sum is then performed over the Becke grid points, and the interaction between points computed using the scattering kernels of Williamson and Zewail [J. C. Williamson and A. H. Zewail, J. Phys. Chem. 1994 , 98 , 2766 ]. These kernels analytically average over the molecular orientations with the cos2 γ selection factor appropriate for one-photon dipole absorption in a perpendicular pump-probe geometry. We show that the method is converged with small grids containing <500 points/atom. We implement the algorithm on a GPU for increased efficiency and emonstrate the algorithm for molecules with up to a few dozen atoms. We explore the accuracy of the independent atom model (IAM) by comparison with our new and more accurate method. We also investigate the possibility of detecting signatures of electronic transitions in polyatomic pump-probe diffraction experiments.

Large-Scale Functional Group Symmetry-Adapted Perturbation Theory on Graphical Processing Units.

Symmetry-adapted perturbation theory (SAPT) is a valuable method for analyzing intermolecular interactions. The functional group SAPT partition (F-SAPT) has been introduced to provide additional insight into the origins of noncovalent interactions. Until now, SAPT analysis has been too costly for large ligand-protein complexes where it could provide key insights for chemical modifications that might improve ligand binding. In this paper, we present a large-scale implementation of a variant of F-SAPT. Two pragmatic choices are made from the outset to render the problem tractable: (1) Ab initio computation of dispersion and exchange-dispersion is replaced with Grimme's empirical dispersion correction. (2) Basis sets with augmented functions are avoided to allow for efficient integral screening. These choices allow the F-SAPT analysis to be written largely in terms of Coulomb and exchange matrix builds which have been implemented efficiently on graphical processing units (GPUs). Our formulation of F-SAPT is routinely applicable to molecules with well over 3000 atoms and 25,000 basis functions and is particularly optimized for the case where one monomer is significantly larger than the other. This is demonstrated explicitly with results from F-SAPT analysis of the full indinavir @ HIV-II protease complex (PDB ID 1HSG ) in a polarized double-ζ basis.

An Ab Initio Exciton Model Including Charge-Transfer Excited States.

The Frenkel exciton model is a useful tool for theoretical studies of multichromophore systems. We recently showed that the exciton model could be used to coarse-grain electronic structure in multichromophoric systems, focusing on singly excited exciton states [ Acc. Chem. Res. 2014 , 47 , 2857 - 2866 ]. However, our previous implementation excluded charge-transfer excited states, which can play an important role in light-harvesting systems and near-infrared optoelectronic materials. Recent studies have also emphasized the significance of charge-transfer in singlet fission, which mediates the coupling between the locally excited states and the multiexcitonic states. In this work, we report on an ab initio exciton model that incorporates charge-transfer excited states and demonstrate that the model provides correct charge-transfer excitation energies and asymptotic behavior. Comparison with TDDFT and EOM-CC2 calculations shows that our exciton model is robust with respect to system size, screening parameter, and different density functionals. Inclusion of charge-transfer excited states makes the exciton model more useful for studies of singly excited states and provides a starting point for future construction of a model that also includes double-exciton states.

Psi4 1.1: An Open-Source Electronic Structure Program Emphasizing Automation, Advanced Libraries, and Interoperability.

Psi4 is an ab initio electronic structure program providing methods such as Hartree-Fock, density functional theory, configuration interaction, and coupled-cluster theory. The 1.1 release represents a major update meant to automate complex tasks, such as geometry optimization using complete-basis-set extrapolation or focal-point methods. Conversion of the top-level code to a Python module means that Psi4 can now be used in complex workflows alongside other Python tools. Several new features have been added with the aid of libraries providing easy access to techniques such as density fitting, Cholesky decomposition, and Laplace denominators. The build system has been completely rewritten to simplify interoperability with independent, reusable software components for quantum chemistry. Finally, a wide range of new theoretical methods and analyses have been added to the code base, including functional-group and open-shell symmetry adapted perturbation theory, density-fitted coupled cluster with frozen natural orbitals, orbital-optimized perturbation and coupled-cluster methods (e.g., OO-MP2 and OO-LCCD), density-fitted multiconfigurational self-consistent field, density cumulant functional theory, algebraic-diagrammatic construction excited states, improvements to the geometry optimizer, and the "X2C" approach to relativistic corrections, among many other improvements.

α-CASSCF: An Efficient, Empirical Correction for SA-CASSCF To Closely Approximate MS-CASPT2 Potential Energy Surfaces.

Because of its computational efficiency, the state-averaged complete active-space self-consistent field (SA-CASSCF) method is commonly employed in nonadiabatic ab initio molecular dynamics. However, SA-CASSCF does not effectively recover dynamical correlation. As a result, there can be qualitative differences between SA-CASSCF potential energy surfaces (PESs) and more accurate reference surfaces computed using multistate complete active space second-order perturbation theory (MS-CASPT2). Here we introduce an empirical correction to SA-CASSCF that scales the splitting between individual states and the state-averaged energy. We call this the α-CASSCF method, and we show here that it significantly improves the accuracy of relative energies and PESs compared with MS-CASPT2 for the chromophores of green fluorescent and photoactive yellow proteins. As such, this method may prove to be quite valuable for nonadiabatic dynamics.